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+arXiv:2301.13134v1  [math.RA]  30 Jan 2023
+The fundamental theorem of calculus in
+differential rings
+Clemens G. Raaba,∗ and Georg Regensburgera,b
+aInstitute for Algebra, Johannes Kepler University Linz, Austria
+bInstitute of Mathematics, University of Kassel, Germany
+clemensr@algebra.uni-linz.ac.at
+regensburger@mathematik.uni-kassel.de
+Abstract
+In this paper, we study the fundamental theorem of calculus and its
+consequences from an algebraic point of view. For functions with singu-
+larities, this leads to a generalized notion of evaluation. We investigate
+properties of such integro-differential rings and discuss many examples.
+We also construct corresponding integro-differential operators and pro-
+vide normal forms via rewrite rules. They are then used to derive several
+identities and properties in a purely algebraic way, generalizing well-known
+results from analysis. In identities like shuffle relations for nested integrals
+and the Taylor formula, additional terms are obtained that take singular-
+ities into account. Another focus lies on treating basics of linear ODEs
+in this framework of integro-differential operators. These operators can
+have matrix coefficients, which allow to treat systems of arbitrary size in a
+unified way. In the appendix, using tensor reduction systems, we give the
+technical details of normal forms and prove them for operators including
+other functionals besides evaluation.
+Keywords
+Integro-differential rings, integro-differential operators, normal
+forms, generalized shuffle relations, generalized Taylor formula
+1
+Introduction
+Differential rings are a well-established algebraic structure for modelling dif-
+ferentiation by derivations, i.e. linear operations satisfying the Leibniz rule.
+More recently, integro-differential algebras have been introduced to additionally
+model integration and point evaluation of continuous univariate functions by
+∗Corresponding author
+1
+
+linear operations satisfying corresponding algebraic identities. In contrast, the
+integro-differential rings introduced in this paper only use the two identities
+d
+dx
+� x
+a
+f(t) dt = f(x)
+and
+� x
+a
+f ′(t) dt = f(x) − f(a)
+of the fundamental theorem of calculus and the Leibniz rule as axioms.
+In
+particular, this results in a generalized notion of evaluation that is only required
+to map to constants. This allows to deal with evaluations of functions even if
+singularities or discontinuities are present. For example, it is natural to consider
+integro-differential rings containing the rational functions leading to so-called
+hyperlogarithms.
+It turns out that many analytic identities and general statements about
+ODEs, like variation of constants, have a purely algebraic proof in integro-
+differential rings that is independent of the analytic properties of concrete
+functions. Likewise, computations with and transformations of linear integro-
+differential equations and initial conditions can be done in this algebraic setting
+as well.
+Moreover, exploring algebraic consequences of the Leibniz rule and
+the fundamental theorem of calculus, we also find new results that introduce
+additional evaluation terms into identities like shuffle relations for nested inte-
+grals and the Taylor formula. In short, we investigate the analytic operations
+of differentiation, integration, and evaluation from a purely algebraic point of
+view. In this context, we implement in some sense the somewhat provocative
+statement of Rota [33, p. 57] that “the algebraic structure sooner or later comes
+to dominate [. . . ]. Algebra dictates the analysis.”
+In order to study linear integro-differential equations and identities, we use
+the operator point of view, which treats identities of functions as identities of
+corresponding linear operators acting on them. To this end, we algebraically
+construct the ring of integro-differential operators. For simplifying expressions
+for operators, we use identities as rewrite rules. As a key result, we work out a
+particular rewrite system that can be applied straightforwardly to obtain nor-
+mal forms and to prove any algebraic identity of integro-differential operators.
+Our construction of the ring of operators can be used for operators with scalar
+coefficients and for operators with matrix coefficients. In particular, it allows to
+uniformly deal with scalar equations as well as with systems, even of undeter-
+mined size. Preliminary versions of some results presented in this paper have
+already been presented by the authors at the conference “Differential Algebra
+and Related Topics” (DART VII) in 2016.
+A related approach was taken in the work of Danuta Przeworska-Rolewicz,
+see for example [23]. In short, she considers a right invertible linear operator on
+a vector subspace as a generalization of derivation. Right inverses and projec-
+tions onto the kernel take the role of integration and evaluation, respectively. In
+this linear setting, she develops algebraic generalizations of results for calculus
+and linear differential equations. She refers to this as algebraic analysis, see
+also [24] for historic context and references. Several results, e.g. a Taylor for-
+mula, can be formulated already in this purely linear setting. For other results,
+multiplication in a commutative algebra is considered. In some statements, the
+2
+
+Leibniz rule or weakened versions of it are needed. However, she does not con-
+sider shuffle relations for nested integrals or additional evaluation terms in the
+Taylor formula. Her treatment of operators is limited to properties and iden-
+tities of given operators and she does not consider rings generated by them or
+normal forms.
+Integro-differential algebras and operators over a field of constants were al-
+ready introduced in [29, 30], see [31] for a detailed overview and further refer-
+ences. More general differential algebras with integration over rings were intro-
+duced in [11], see [12] for a unified presentation and comparison. In contrast, the
+integro-differential rings defined in [14, 13] require the integration to be linear
+over all constants, but the construction of corresponding operators introduced
+there allows noncommutative coefficients and constants. Further references to
+the literature can be found in the respective sections of the present paper.
+So far, all algebraic treatments of integro-differential operators in the litera-
+ture restrict to multiplicative evaluations, i.e. evaluation of a product is the prod-
+uct of the individual evaluations. Assuming only the Leibniz rule and the iden-
+tities of the fundamental theorem of calculus, we deal with non-multiplicative
+evaluations quite naturally in this paper. In Section 2, we introduce (general-
+ized) integro-differential rings following this principle. Analyzing the relations
+imposed, we show for example that any linear projection onto constants may be
+used as the evaluation of such an integro-differential ring. We also present many
+examples, both with multiplicative and with non-multiplicative evaluation. Re-
+mark 2.12 discusses the differences among our definition and the definitions in
+the literature.
+In Section 3, we investigate identities satisfied by integrals and their prod-
+ucts. This reveals a generalization of the Rota-Baxter identity for integration
+that contains an additional evaluation term. Focusing on nested integrals, we
+discover generalized shuffle relations with elaborate additional terms involving
+nested integrals of lower depth. We also characterize properties of repeated in-
+tegrals of 1, which form the smallest integro-differential ring with given ring of
+constants.
+Linear integro-differential operators with coefficients in arbitrary (gener-
+alized) integro-differential rings are constructed algebraically in Section 4 by
+generators and relations. Working at the operator level enables statements of
+broader applicability, since operators not only act on integro-differential rings
+but also on more general modules. We compute a complete set of rewrite rules
+to simplify such operators to normal form. A precise analysis of the uniqueness
+of these normal forms is presented in the appendix only, since it requires a re-
+fined construction relying on tensor rings (like in [14, 13] for the multiplicative
+case). After a largely self-contained introduction to tensor reduction systems in
+the appendix, this is carried out in Section A.4 allowing also other functionals
+besides evaluation. In Section 4.1, we collect properties of integro-differential
+operators with coefficients from an integral domain (e.g. analytic functions). In
+particular, we also characterize the action of such operators when evaluation is
+multiplicative.
+In the remaining sections, we illustrate how computations in the ring of
+3
+
+integro-differential operators can be used to prove and generalize well-known
+results from analysis. For example, variation of constants remains valid for ar-
+bitrary integro-differential rings, which is the focus of Section 5. In particular,
+we also detail how integro-differential operators with noncommutative coeffi-
+cients can be used for proving statements about systems of arbitrary or even
+undetermined size. In Section 6, we discuss how results from analysis need to
+be modified for allowing the induced evaluation to be non-multiplicative. First,
+we look at the formula for variation of constants, which for multiplicative eval-
+uations automatically satisfies homogeneous initial conditions, and include an
+extra term to retain this property in general. Finally, we present a version of the
+Taylor formula with integral remainder term that is valid also for generalized
+evaluations.
+Conventions
+Throughout the paper, rings are implicitly assumed to have
+a unit element and to be different from the zero ring.
+Unless stated other-
+wise, rings are not assumed to be commutative and can be of arbitrary char-
+acteristic. Nevertheless, for easier reading, we use the notions of modules and
+linear maps from the commutative setting to refer to bimodules and bimodule-
+homomorphisms over noncommutative rings. In addition, we use operator no-
+tation for linear maps, e.g. the Leibniz rule for the derivative of products then
+reads ∂fg = (∂f)g + f∂g.
+2
+Integro-differential rings
+To uniformly deal with differentiation of various kinds of functions, we use a
+few basic abstract notions. Recall from differential algebra that a derivation on
+a ring R is an additive map ∂ : R → R that satisfies the Leibniz rule
+∂fg = (∂f)g + f∂g
+(1)
+for all f, g ∈ R. Then, (R, ∂) is called a differential ring and f ∈ R is called
+a constant in this differential ring if and only if ∂f = 0. It is easy to see that
+the set of constants forms a subring of R and ∂ is linear w.r.t. the ring of its
+constants.
+For further theory of differential rings see e.g. [15].
+In addition,
+we introduce the following notions of integration and evaluation in differential
+rings.
+Definition 2.1. Let (R, ∂) be a differential ring and let C be its ring of con-
+stants. We call a C-linear map
+�
+: R → R an integration on R, if
+∂
+�
+f = f
+(2)
+holds for all f ∈ R. A C-linear functional e: R → C which acts on C as the
+identity is called an evaluation on R.
+In other words, integrations on differential rings are right inverses of the
+derivation that are linear over the constants and evaluations on differential rings
+are C-linear projectors onto the ring of constants C.
+4
+
+Definition 2.2. Let (R, ∂) be a differential ring and let
+�
+: R → R be an
+integration on R. We call (R, ∂,
+�
+) a (generalized) integro-differential ring and
+we define the (induced) evaluation E on R by
+Ef := f −
+�
+∂f.
+(3)
+If in addition R is a field or skew field, then we also call (R, ∂,
+�
+) a (generalized)
+integro-differential (skew) field, respectively.
+This extends the definition of integro-differential rings in [14] by dropping
+the additional requirement that the induced evaluation should be multiplicative.
+In the present paper, the notion of integro-differential rings always refers to
+Definition 2.2. The following lemma shows that in any integro-differential ring,
+the (induced) evaluation E is indeed an evaluation as defined in Definition 2.1.
+Moreover, the ring R can be decomposed as direct sum of constant and non-
+constant “functions”.
+Lemma 2.3. Let (R, ∂,
+�
+) be an integro-differential ring with constants C.
+Then, for all f ∈ R and c ∈ C, we have Ef ∈ C, E
+�
+f = 0, and Ec = c.
+Moreover,
+R = C ⊕
+�
+R
+as direct sum of C-modules.
+Proof. First, we compute ∂Ef = ∂(f − � ∂f) = ∂f − ∂f = 0 and E� f =
+�
+f −
+�
+∂
+�
+f = 0 for f ∈ R as well as Ec = c−
+�
+∂c = c for c ∈ C. For any f ∈ R,
+we have f = Ef + f − Ef = Ef +
+�
+∂f and hence R = C +
+�
+R. Let f ∈ C ∩
+�
+R
+and g ∈ R such that f =
+�
+g. Then, 0 = ∂f = ∂
+�
+g = g, which implies f = 0.
+Hence, the sum R = C +
+�
+R is direct.
+By the previous lemma, any integration induces an evaluation by id −
+�
+∂.
+Conversely, any evaluation e can be used to define an integration that has e as
+its induced evaluation, as the following theorem shows. It is easy to see that
+two different integrations cannot have the same induced evaluation. Altogether,
+on differential rings with ∂R = R, there is a one-to-one correspondence of
+integrations and evaluations.
+Theorem 2.4. Let (R, ∂) be a differential ring such that ∂R = R and let e be
+an evaluation on R. Define
+�
+e : R → R by
+�
+ef := g − eg
+for all f ∈ R, where g ∈ R is such that ∂g = f. Then (R, ∂,
+�
+e) is an integro-
+differential ring and the induced evaluation is E = e.
+Moreover, any integration
+�
+on R can be obtained from its induced evaluation
+E via this construction:
+�
+=
+�
+E.
+Proof. Let C be the ring of constants of (R, ∂). First, we show that
+�
+e is well-
+defined.
+If g, ˜g ∈ R are such that ∂g = ∂˜g, then with c := ˜g − g ∈ C we
+5
+
+have ˜g − e˜g = g + c − eg − ec = g − eg, since ec = c by definition of e. For
+showing C-linearity of
+�
+e, we let c ∈ C and f1, f2, g1, g2 ∈ R with ∂gi = fi.
+Then, ∂(cg1 + g2) = cf1 + f2 together with C-linearity of id − e implies C-
+linearity of
+�
+e.
+Consequently, (R, ∂,
+�
+e) is an integro-differential ring, since
+we also have ∂
+�
+ef = f by construction. The induced evaluation is given by
+Ef = f −
+�
+e∂f = f − (f − ef) = ef for f ∈ R.
+Let
+�
+: R → R be any C-linear right inverse of ∂, E its induced evaluation,
+and f ∈ R. Then, we have �
+Ef = � f − E� f = � f by definition of �
+E and by
+Lemma 2.3.
+In particular, if (R, ∂,
+�
+) is an integro-differential ring and e is any evaluation
+on R, then the integration that induces e can be given in terms of
+�
+by
+�
+e :=
+�
+− e
+�
+.
+(4)
+This implies that the difference of two integrations
+�
+1,
+�
+2 on the same differential
+ring can be given as
+�
+1−
+��
+2 = E2
+�
+1 = −E1
+�
+2 in terms of the induced evaluations
+E1, E2. More generally, if (R, ∂,
+�
+) is an integro-differential ring with constants
+C and e : R → C is only C-linear, then
+�
+e defined by (4) can be easily seen to be
+an integration on R and its induced evaluation is given by e + (id − e)E, which
+agrees with e if and only if the latter is an evaluation (i.e. e1 = 1).
+By Lemma 2.3,
+�
+R is a direct complement of C in an integro-differential ring
+R. Conversely, any direct complement of C gives rise to an evaluation on R,
+which in turn induces an integration by Theorem 2.4. More precisely, we have
+the following characterization of integro-differential rings.
+Corollary 2.5. Let (R, ∂) be a differential ring with ring of constants C. Then,
+(R, ∂) can be enriched into an integro-differential ring if and only if ∂R = R and
+C is a complemented C-module in R. Moreover, if ∂R = R, there exists a one-
+to-one correspondence between direct complements of C in R and integrations
+on (R, ∂).
+This characterization shows that on an integro-differential ring (R, ∂,
+�
+),
+in general, there are many other integrations that make R into an integro-
+differential ring with the same derivation. In contrast, the following character-
+ization shows that, in general, ∂ is the only derivation that turns R into an
+integro-differential ring with the same integration.
+Lemma 2.6. Let R be a ring, let C be a subring of R, and let
+�
+: R → R be a
+C-linear map. Then, there exists a derivation ∂ on R such that (R, ∂, � ) is an
+integro-differential ring with constants C if and only if the following conditions
+hold.
+1.
+�
+is injective.
+2. R = C ⊕
+�
+R
+3. (
+�
+f)
+�
+g −
+�
+(
+�
+f)g −
+�
+f
+�
+g ∈ C for all f, g ∈ R.
+6
+
+Moreover, this derivation is unique if it exists.
+Proof. First, it is easy to see, from injectivity of
+�
+and by R = C ⊕
+�
+R, that
+there exists a C-linear map ∂ : R → R such that ker ∂ = C and ∂
+�
+= id and
+that this map is unique. To show that ∂ is indeed a derivation, we verify the
+Leibniz rule on two arbitrary elements of R. By R = C ⊕
+�
+R, we write these
+two elements as c+
+�
+f and d+
+�
+g with c, d ∈ C and f, g ∈ R. Now, we compute
+∂(c + � f)(d + � g) − (∂(c + � f))(d + � g) − (c + � f)∂(d + � g)
+= ∂(
+�
+f)
+�
+g − f
+�
+g − (
+�
+f)g
+= ∂
+�
+(
+�
+f)
+�
+g −
+�
+(
+�
+f)g −
+�
+f
+�
+g
+�
+,
+which is zero by ker ∂ = C and the last assumption on
+�
+.
+Conversely, if (R, ∂,
+�
+) is an integro-differential ring with constants C, then
+injectivity of
+�
+follows from the definition (2) and the other two conditions on
+�
+follow from Lemma 2.3 and Theorem 3.1.
+It is straightforward to equip the matrix ring over an integro-differential
+ring with an integro-differential ring structure. Such noncommutative integro-
+differential rings are relevant when working with linear systems, see Section 5.2.
+Lemma 2.7. Let (R, ∂,
+�
+) be an integro-differential ring with constants C and
+let n ≥ 1.
+Then, (Rn×n, ∂, � ) is an integro-differential ring with constants
+Cn×n, where operations ∂,
+�
+, E act on matrices by applying the corresponding
+operation entrywise in R.
+Proof. Clearly, ∂,
+�
+, E are Cn×n-linear on Rn×n satisfying ∂
+�
+A = A and
+�
+∂A =
+A − EA for all A ∈ Rn×n. Moreover, it is straightforward to verify that ∂ is a
+derivation on R with constants Cn×n.
+Example 2.8. Basic examples for commutative integro-differential rings are
+univariate polynomials C[x] and formal power series C[[x]] over a commutative
+ring C with Q ⊆ C. The derivation is given by ∂ =
+d
+dx with ring of constants C
+and integration is defined C-linearly by
+�
+xn = xn+1
+n + 1
+for all n ∈ N.
+The induced evaluation extracts the constant coefficient and
+corresponds to evaluation at 0.
+If C ⊆ C, the integration
+�
+corresponds to
+integration
+� x
+0 from 0. Also the rings of complex-valued smooth or analytic
+functions on a (possibly unbounded) interval I ⊆ R together with derivation
+∂ =
+d
+dx and integration
+�
+=
+� x
+a , for fixed a ∈ I, are integro-differential rings.
+Then, the induced evaluation is the evaluation of functions at the point a.
+In particular, the ring of exponential polynomials on the real line generated
+by polynomials and exponential functions is closed under differentiation and
+integration and hence is an integro-differential ring as well. Algebraically, for any
+field C of characteristic zero, we can consider the ring of exponential polynomials
+7
+
+C[x, eCx], where ecxedx = e(c+d)x for all c, d ∈ C and e0x = 1, together with
+derivation ∂ =
+d
+dx and with the integration that is induced by evaluation at 0
+based on Theorem 2.4.
+Example 2.9. A basic example of integro-differential rings of arbitrary charac-
+teristic are Hurwitz series, which are closely related to formal power series and
+have been defined in [16, 17], with derivation ∂(a0, a1, . . . ) = (a1, a2, . . . ) and
+integration given by
+�
+(a0, a1, . . . ) = (0, a0, a1, . . . ), see also [18].
+The examples mentioned so far have the special property that the evalua-
+tion of the integro-differential ring is multiplicative, as in the usual definition of
+integro-differential algebras. However, for certain differential rings (in particu-
+lar for differential fields, cf. Corollary 5 in [31]), it is not possible to define a
+multiplicative evaluation for the following reason.
+Remark 2.10. If the induced evaluation is multiplicative, one can see easily
+that no element of
+�
+R can have a multiplicative inverse. Since otherwise we
+would have Ef 1
+f = E1 = 1 and (Ef)E 1
+f = 0E 1
+f = 0 for such f ∈ � R.
+Analytically, if evaluation should correspond to evaluation of functions at a
+fixed point a for functions that are continuous at a, then requiring multiplica-
+tivity of evaluation means that functions with poles at a cannot be considered.
+In particular, if a function f has a pole of order m at a, then evaluation of the
+product (x − a)mf gives a nonzero value, but the factor (x − a)m evaluates to
+zero at a.
+In the following theorem, based on results from the literature, we briefly
+characterize when the induced evaluation is multiplicative. From (3) it imme-
+diately follows that the identity (7) is equivalent to multiplicativity of E. Since
+in integro-differential rings
+�
+is C-linear by definition, we have the following
+characterization of integro-differential rings with multiplicative evaluation.
+Theorem 2.11. Let (R, ∂,
+�
+) be an integro-differential ring. Then the following
+properties are equivalent.
+1. E is multiplicative, i.e. for all f, g ∈ R we have
+Efg = (Ef)Eg.
+(5)
+2.
+�
+satisfies the Rota-Baxter identity, i.e. for all f, g ∈ R we have
+(� f)� g = � (� f)g + � f� g.
+(6)
+3. The hybrid Rota-Baxter identity holds, i.e. for all f, g ∈ R we have
+(
+�
+∂f)
+�
+∂g = (
+�
+∂f)g + f
+�
+∂g −
+�
+∂fg.
+(7)
+Proof. Since (R, ∂) is a differential Z-algebra of weight 0 and
+�
+is Z-linear, this
+immediately follows from items (b), (g), and (a) of Theorem 2.5 in [12].
+8
+
+Remark 2.12. In the literature, integro-differential K-algebras (of weight 0)
+over a commutative ring K with unit element are defined as differential K-
+algebras where the additional map
+�
+is only required to be K-linear, but has
+to satisfy the hybrid Rota-Baxter axiom (7) in addition to (2), see [12]. Analo-
+gously, the more general notion of differential Rota-Baxter K-algebras (of weight
+0) imposes the Rota-Baxter identity (6) instead of (7) in addition to (2). On a
+differential Rota-Baxter K-algebra with constants C, a K-linear map E is defined
+by (3) as well and, by Theorem 2.5 in [12], properties (5), (7), and C-linearity
+of
+�
+are equivalent, see also Proposition 10 in [31].
+By the previous theorem, any (generalized) integro-differential ring with
+constants C is an integro-differential K-algebra for K = Z and for K = C ∩Z(R)
+(i.e. K = C, if R is commutative) if any of the equivalent conditions holds.
+Conversely, any differential Rota-Baxter K-algebra (of weight 0) is an integro-
+differential ring if and only if
+�
+is linear over the constants C. In particular,
+this is automatically the case for integro-differential K-algebras (of weight 0) by
+Proposition 10 in [31]. For concrete differential Rota-Baxter algebras (of weight
+0) where
+�
+is not C-linear, see Example 3 in [30] and the algebraic analog of
+piecewise functions constructed in [32].
+As a basic example for integro-differential rings with non-multiplicative eval-
+uation, we extend the polynomial ring C[x] over a commutative ring C with
+Q ⊆ C by adjoining the multiplicative inverse x−1. In order to have a surjective
+derivation ∂ =
+d
+dx, we also need to adjoin the logarithm ln(x) as in the following
+example.
+Example 2.13. On C[x, x−1, ln(x)], with Q ⊆ C and ∂ =
+d
+dx, we can define the
+C-linear integration recursively as follows.
+� xk ln(x)n :=
+
+
+
+
+
+
+
+xk+1
+k+1
+k ̸= −1 ∧ n = 0
+xk+1
+k+1 ln(x)n −
+n
+k+1
+�
+xk ln(x)n−1
+k ̸= −1 ∧ n > 0
+ln(x)n+1
+n+1
+k = −1
+The same recursive definition also works on the larger ring C((x))[ln(x)] of formal
+Laurent series with logarithms, where every element can be written in the form
+�∞
+k=−m
+�m
+n=0 ck,nxk ln(x)n for some m ∈ N and ck,n ∈ C. In both cases, the
+induced evaluation acts by
+E
+∞
+�
+k=−m
+m
+�
+n=0
+ck,nxk ln(x)n = c0,0
+and is not multiplicative as expected by Remark 2.10. Moreover, for the integro-
+differential subrings of polynomials or formal power series, this evaluation cor-
+responds to the usual multiplicative evaluation at 0.
+Example 2.14. Rational functions together with nested integrals of rational
+functions also form an integro-differential ring with non-multiplicative evalua-
+tion. Algebraically, if C is a field of characteristic zero, this can be understood
+9
+
+as an integro-differential subring of C((x))[ln(x)] with integration
+�
+as in the
+previous example. In fact, this ring is the smallest integro-differential ring con-
+taining C(x) and is generated as a C(x)-vector space by 1 and all nested inte-
+grals
+�
+f1
+�
+f2
+�
+. . .
+�
+fn of arbitrary depth n ≥ 1, where fi ∈ C(x) are proper
+and have irreducible denominators. In particular, if C = C, the integrands can
+be chosen as fi =
+1
+x−ai with ai ∈ C. These kind of nested integrals are called
+hyperlogarithms [21] and have been investigated already in [20]. In [12], the
+free integro-differential algebra (having multiplicative evaluation) generated by
+C(x) has been constructed at the somewhat unnatural expense that
+�
+1 ̸∈ C(x)
+and the constants of the resulting differential ring contain much more than just
+C.
+Example 2.15. Another example of an integro-differential ring that contains
+the rational functions are the D-finite functions [35]. They are characterized
+as solutions of linear differential equations with rational function coefficients
+and indeed form a differential ring with surjective derivation
+d
+dx.
+Since the
+constants of this differential ring are given by C, there exists an integration by
+Corollary 2.5.
+Example 2.16. All the examples considered above are just rings, not fields.
+In contrast, transseries R[[[x]]] are an explicit construction of a differential
+field (with field of constants R) that is closed under taking antiderivatives,
+see [36, 8, 1] and references therein.
+Thus, R[[[x]]] can be turned into an
+integro-differential field by Corollary 2.5 whose evaluation necessarily is non-
+multiplicative by Remark 2.10.
+As shown by Corollary 2.5, in general, there are many different choices for
+an integration
+�
+in order to turn a differential ring with ∂R = R into an
+integro-differential ring. On the same differential ring, for some integrations
+the induced evaluation is multiplicative and for others it is not. It may even
+be the case that a canonical choice of
+�
+yields a non-multiplicative evaluation
+while there are other choices that would give a multiplicative evaluation. In
+particular, this is the case for exponential polynomials, for example. They were
+mentioned above with a multiplicative evaluation, while the following canonical
+definition of
+�
+gives rise to a non-multiplicative one.
+Example 2.17. The ring of exponential polynomials C[x, eCx] over a field C of
+characteristic zero is C-linearly generated by terms of the form xkecx with k ∈ N
+and c ∈ C. Apart from the evaluation-based integration on C[x, eCx] mentioned
+above, it is quite natural to define a C-linear integration
+�
+recursively as follows.
+�
+xkecx :=
+
+
+
+
+
+
+
+xk+1
+k+1
+c = 0
+1
+cecx
+k = 0 ∧ c ̸= 0
+1
+cxkecx − k
+c
+�
+xk−1ecx
+k > 0 ∧ c ̸= 0
+In terms of the Pochhammer symbol (a)k := a·(a + 1)· . . . ·(a + k − 1),
+�
+can be
+10
+
+given explicitly as
+�
+xkecx =
+k
+�
+i=0
+(−k)k−ici−k−1xiecx.
+Then, the induced evaluation Ef := f −
+�
+∂f acts by
+Exkecx =
+�
+1
+k = c = 0
+0
+otherwise
+and is not multiplicative since we have Eecxe−cx = E1 = 1 but Ee±cx = 0 for any
+c ̸= 0, for example. On the other hand, C[x, ex] is an integro-differential subring
+with multiplicative evaluation, for example. With this integration, for instance,
+the subset C[x, ex]ex is closed under addition, multiplication, derivation, and
+integration and, hence, could be viewed as an integro-differential subring with-
+out unit element and having multiplicative evaluation and the zero ring as its
+constants.
+All the examples with explicit integration discussed so far contain an integro-
+differential subring on which the induced evaluation is multiplicative. In general,
+however, this need not be the case as the following example shows.
+Example 2.18. On C[x] with the usual derivation and Q ⊆ C, for example,
+we can define a C-linear integration by
+�
+xn = xn+1
+n+1 + c for all n ∈ N for any
+fixed c ∈ Z(C). Such an integration induces the evaluation Ef = f(0) − cf ′(1)
+on C[x], which is not multiplicative if c ̸= 0 (e.g. f = x and g = x2 − 2x yield
+Eg = 0 and Efg = c).
+3
+Products of nested integrals
+In integro-differential rings with multiplicative evaluation the standard Rota-
+Baxter identity (6) allows to write the product of integrals as a sum of two
+nested integrals. For nested integrals, this leads to shuffle identities [27] where
+a product of two nested integrals is expressed as a sum of nested integrals.
+More generally, for Rota-Baxter operators with weight and corresponding shuffle
+products involving additional terms, see [10] and references therein. In general
+integro-differential rings, i.e. if E is not multiplicative, additional terms involving
+the evaluation arise in the identities (6) and (7) and also in the shuffle identi-
+ties. Note that all evaluation terms are evaluations of products of integrals.
+Therefore, they vanish if E is multiplicative, since E� f = 0 for all f ∈ R.
+Theorem 3.1. Let (R, ∂,
+�
+) be an integro-differential ring. Then the Rota-
+Baxter identity with evaluation
+(
+�
+f)
+�
+g =
+�
+f
+�
+g +
+�
+(
+�
+f)g + E(
+�
+f)
+�
+g
+(8)
+holds for all f, g ∈ R as well as
+(
+�
+∂f)
+�
+∂g = (
+�
+∂f)g + f
+�
+∂g −
+�
+∂fg − E(
+�
+∂f)
+�
+∂g.
+(9)
+11
+
+Proof. Using (3), we can effect the decomposition of R shown in Lemma 2.3.
+For (
+�
+f)
+�
+g, we thereby obtain the decomposition
+�
+∂(
+�
+f)
+�
+g + E(
+�
+f)
+�
+g, which
+implies (8) by the Leibniz rule for the derivation ∂. By
+�
+∂f = f − Ef,
+�
+∂g =
+g − Eg, and
+�
+∂fg = fg − Efg, one can write (9) in a form that can be easily
+verified using the fact that E is an evaluation.
+As a first application, we show that in every integro-differential ring the re-
+peated integrals of 1 give rise to an integro-differential subring. It is the smallest
+integro-differential ring with the same constants. In characteristic zero, this ring
+consists of the univariate polynomials with coefficients in the constants and, for
+nonzero characteristic, it consists of a finite version of Hurwitz series [17].
+Theorem 3.2. Let (R, ∂,
+�
+) be an integro-differential ring with constants C.
+For all n ≥ 1, let xn :=
+� n1 ∈ R and let x0 := 1. Then, 1, x1, x2, . . . commute
+with all elements of C and are C-linearly independent. The C-module
+P := spanC{1, x1, x2, . . .}
+is an integro-differential subring of R. If Q ⊆ R, then P = C[x1].
+Moreover, E is multiplicative on P if and only if Exmxn = 0 for m, n ≥ 1.
+Assuming E is multiplicative on P, then we have xmxn =
+�m+n
+m
+�
+xm+n and, if
+in addition Q ⊆ R, xn = 1
+n!xn
+1 for m, n ∈ N.
+Proof. Since
+�
+is C-linear, every element of C commutes with xi for every i ∈ N,
+even if R or C is noncommutative.
+So, every element of P is of the form
+�n
+i=0 cixi, for some ci ∈ C. To show C-linear independence of x0, x1, . . . , let n ∈
+N be minimal such that there are c0, . . . , cn ∈ C with cn ̸= 0 and �n
+i=0 cixi = 0.
+Then, �n−1
+i=0 ci+1xi = ∂ �n
+i=0 cixi = 0 would imply n = 0 by minimality of
+n. Because this would yield c0 = 0, we conclude that x0, x1, . . . are C-linearly
+independent.
+Obviously, P is closed under ∂ and
+�
+since both operations are C-linear with
+∂xn ∈ P and
+�
+xn = xn+1 for all n ∈ N. For showing that P is closed under
+multiplication, it suffices to show that xmxn ∈ P for all m, n ≥ 1. We proceed
+by induction on the sum n + m. For m = n = 1 we have x2
+1 = 2x2 + Ex2
+1 by (8).
+For m + n > 2 we have
+xmxn =
+�
+xm−1xn +
+�
+xmxn−1 + Exmxn
+by (8). By the induction hypothesis, xm−1xn and xmxn−1 are in P. Hence, the
+same is also true after applying
+�
+, which completes the induction. Altogether,
+P is an integro-differential subring of R.
+For showing P = C[x1], it is sufficient to prove that every xn is contained
+in C[x1]. By (8), we obtain xn+1 = x1xn −
+�
+xn−1x1 − Ex1xn for all n ≥ 1.
+Therefore, xn ∈ C[x1] follows by induction, if C[x1] is closed under
+�
+. Assuming
+Q ⊆ R, we verify that
+�
+xn
+1 −
+1
+n+1xn+1
+1
+∈ C for all n ≥ 1 by applying ∂ to it,
+which shows
+�
+xn
+1 ∈ C[x1] for all n ≥ 1.
+Moreover, any xn with n ≥ 1 satisfies Exn = 0. So, if E is multiplicative on
+P, then trivially Exmxn = (Exm)Exn = 0 for m, n ≥ 1. Conversely, since P
+12
+
+is generated by 1, x1, x2, . . . as a C-module, we know that
+�
+P is generated by
+x1, x2, . . . as a C-module. Hence, applying E to the product of two elements of
+�
+P gives 0, if Exmxn = 0 for all m, n ≥ 1. Altogether, using the decomposition
+P = C ⊕
+�
+P given by Lemma 2.3, we conclude that E is multiplicative on P, if
+Exmxn = 0 for all m, n ≥ 1.
+Now, we assume E is multiplicative on P and let m, n ∈ N. If m + n ≤ 1,
+then xmxn =
+�m+n
+m
+�
+xm+n and xn =
+1
+n!xn
+1 hold trivially. For m + n ≥ 2, it fol-
+lows inductively by (8) that xmxn = � �m+n−1
+m−1
+�
+xm+n−1 + � �m+n−1
+m
+�
+xm+n−1 =
+�m+n
+m
+�
+xm+n. In particular, for n ≥ 2, x1xn−1 = nxn implies xn =
+1
+n!xn
+1 induc-
+tively, if Q ⊆ R.
+Even if E is not multiplicative on P, we can analyze some properties of the
+sequence of constants cm,n := Exmxn with n, m ≥ 1. By C-linearity of � and
+E, it trivially follows that all cm,n are in Z(C). It can be shown that all xn
+commute with each other w.r.t. multiplication if and only if the sequence cm,n
+is symmetric.
+If Q ⊆ R, it can be shown by lengthy computation that the
+constants cm,n are determined by all c1,n via the recursion
+cm,n = 1
+m
+��m+n−1
+m−1
+�
+c1,m+n−1 +
+m−2
+�
+j=0
+n−1
+�
+k=1
+�j+k
+j
+�
+c1,j+kcm−j−1,n−k
+�
+.
+To express xn in terms of powers of x1 in general, for Q ⊆ R, we obtain the
+recursion xn = 1
+n!xn
+1 − �n
+i=2
+1
+i!xn−iExi
+1 from Theorem 6.2.2 in [23].
+3.1
+Generalized shuffle relations
+In this section, we let (R, ∂,
+�
+) be a commutative integro-differential ring. Iter-
+ating the standard Rota-Baxter identity (6) leads to shuffle relations for nested
+integrals expressing a product of two nested integrals of depth m and n as a
+sum of nested integrals of depth exactly m + n. Also by recursively applying
+the Rota-Baxter identity with evaluation (8), products of nested integrals can
+be rewritten in R as sums of nested integrals where also terms of lower depth
+may occur. For convenient notation of the formulae involved, it is standard to
+work in tensor products of R and to use the shuffle product, which we recall in
+the following (see [10] for example).
+We consider the C-module C⟨R⟩ := �∞
+n=0 R⊗n, where the tensor prod-
+uct is taken over C and the empty tensor is denoted by ε. Let pure tensors
+a1⊗ . . . ⊗an ∈ C⟨R⟩ represent nested integrals
+�
+a1
+�
+a2 . . .
+�
+an ∈ R. More for-
+mally, by C-linearity of
+�
+, we consider the unique C-module homomorphism
+ϕ : C⟨R⟩ → R such that
+ϕ(a1⊗ . . . ⊗an) =
+�
+a1
+�
+a2 . . .
+�
+an ∈ R
+and ϕ(ε) = 1 ∈ R. For pure tensors a ∈ C⟨R⟩, we denote shortened versions
+of them by aj
+i := ai⊗ai+1⊗ . . . ⊗aj, where aj
+i := ε ∈ R⊗0 if i = j + 1. The
+13
+
+shuffle product on C⟨R⟩ can be recursively defined as follows. For pure tensors
+a, b ∈ C⟨R⟩ of length m and n, respectively, we set
+a
+� b :=
+�
+a ⊗ b
+if m = 0 ∨ n = 0
+a1 ⊗ (am
+2
+� b) + b1 ⊗ (a
+� bn
+2)
+otherwise
+in R⊗(m+n). Extending this definition to C⟨R⟩ by C-linearity, the shuffle product
+turns C⟨R⟩ into a commutative C-algebra.
+Using the shuffle product for pure tensors, the product of nested integrals in
+R can now be represented as sum of nested integrals as follows. The constant
+coefficients of nested integrals of lower depth are evaluations of products of
+integrals. Consequently, if E is multiplicative, then we recover the standard
+shuffle relations [27] with all these constant coefficients equal zero.
+Theorem 3.3. Let (R, ∂,
+�
+) be a commutative integro-differential ring with
+constants C. Let f, g ∈ C⟨R⟩ be pure tensors of length m and n, respectively.
+Then, the product of the nested integrals ϕ(f) =
+�
+f1
+�
+f2 . . .
+�
+fm and ϕ(g) =
+�
+g1
+�
+g2 . . .
+�
+gn is given by
+ϕ(f)ϕ(g) = ϕ(f
+� g) +
+m−1
+�
+i=0
+n−1
+�
+j=0
+e(f m
+i+1, gn
+j+1)ϕ(f i
+1
+� gj
+1) ∈ R
+(10)
+with constants e(f m
+i+1, gn
+j+1) := Eϕ(f m
+i+1)ϕ(gn
+j+1) ∈ C.
+Proof. Without loss of generality, assume m ≤ n. We proceed by induction
+on m. If m = 0, then f = cε for some c ∈ C and the equation (10) reads
+cϕ(g) = ϕ(cε ⊗ g), which is trivially true since cε ⊗ g = cg and ϕ is C-linear.
+For m ≥ 1, we proceed by induction on n. By virtue of (8), for n ≥ m, we have
+ϕ(f)ϕ(g) = � f1ϕ(f m
+2 )ϕ(g) + � ϕ(f)g1ϕ(gn
+2 ) + e(f, g). The product ϕ(f m
+2 )ϕ(g)
+is covered by the induction hypothesis on m so that we obtain
+�
+f1ϕ(f m
+2 )ϕ(g) =
+�
+f1ϕ(f m
+2
+� g) +
+m−1
+�
+i=1
+n−1
+�
+j=0
+e(f m
+i+1, gn
+j+1)
+�
+f1ϕ(f i
+2
+� gj
+1)
+by (10). The product ϕ(f)ϕ(gn
+2 ) is covered by the induction hypothesis on n
+(or on m, if n = m) so that (10) yields
+� ϕ(f)g1ϕ(gn
+2 ) = � g1ϕ(f
+� gn
+2 ) +
+m−1
+�
+i=0
+n−1
+�
+j=1
+e(f m
+i+1, gn
+j+1)� g1ϕ(f i
+1
+� gj
+2).
+By definition of ϕ and
+�, we have
+�
+f1ϕ(f m
+2
+� g) +
+�
+g1ϕ(f
+� gn
+2 ) = ϕ(f
+� g)
+and similarly
+�
+f1ϕ(f i
+2
+� gj
+1) +
+�
+g1ϕ(f i
+1
+� gj
+2) = ϕ(f i
+1
+� gj
+1) for i, j ≥ 1 as well
+as
+�
+f1ϕ(f i
+2
+� g0
+1) = ϕ(f m
+1
+� g0
+1) and
+�
+g1ϕ(f 0
+1
+� gj
+2) = ϕ(f 0
+1
+� gj
+1). Altogether,
+14
+
+this yields
+ϕ(f)ϕ(g) = ϕ(f
+� g) +
+m−1
+�
+i=1
+n−1
+�
+j=1
+e(f m
+i+1, gn
+j+1)ϕ(f i
+1
+� gj
+1)
++
+m−1
+�
+i=1
+e(f m
+i+1, gn
+1 )ϕ(f m
+1
+� g0
+1) +
+n−1
+�
+j=1
+e(f m
+i+1, gn
+j+1)ϕ(f 0
+1
+� gj
+1) + e(f, g),
+which proves (10).
+4
+Integro-differential operators
+In the following, starting from a given integro-differential ring (R, ∂,
+�
+), we
+define the corresponding ring of operators by generators ∂,
+�
+, E and relations.
+As additive maps on R, any f ∈ R acts as multiplication operator g �→ fg and
+satisfies certain identities together with the maps ∂,
+�
+, E. Those identities of
+additive maps that correspond to the defining properties of the operations on
+R will be used as defining relations for the abstract ring of operators below.
+In particular, the Leibniz rule ∂fg = f∂g + (∂f)g of the derivation ∂ on R
+implies the identity ∂ ◦ f = f ◦ ∂ + ∂f of additive maps for every multiplication
+operator f ∈ R. This motivates the identity (11) in the definition below. Sim-
+ilarly, the identities (2) and (3) in R give rise to the identities (12) and (13).
+Moreover, from C-linearity of the operations ∂,
+�
+, E we obtain
+�
+cg = c
+�
+g for all
+c ∈ C and g ∈ R, for example. In addition, we also obtain
+�
+fEg = (
+�
+f)Eg for
+all f, g ∈ R, since Eg ∈ C. Hence, we also impose commutativity of ∂,
+�
+, E with
+elements of C and the identities (14)–(16) in the following definition.
+Definition 4.1. Let (R, ∂,
+�
+) be an integro-differential ring and let C be its ring
+of constants. We let
+R⟨∂,
+�
+, E⟩
+be the (noncommutative unital) ring extension of R generated by indeterminates
+∂,
+�
+, E, where ∂,
+�
+, E commute with all elements of C and the following identities
+hold for all f ∈ R.
+∂ · f = f · ∂ + ∂f
+(11)
+∂ ·
+�
+= 1
+(12)
+�
+· ∂ = 1 − E
+(13)
+∂ · f · E = ∂f · E
+(14)
+�
+· f · E =
+�
+f · E
+(15)
+E · f · E = Ef · E
+(16)
+We call R⟨∂,
+�
+, E⟩ the ring of (generalized) integro-differential operators (IDO).
+Note that, for multiplication in R⟨∂,
+�
+, E⟩, we always explicitly write · when
+one of ∂,
+�
+, E is involved. This is necessary in order to distinguish the product
+15
+
+∂ · f of operators from the multiplication operator ∂f, for example. By con-
+struction, the ring R⟨∂,
+�
+, E⟩ has a natural action on R, where the elements of
+R act as multiplication operators and ∂,
+�
+, E act as the corresponding opera-
+tions. With this action, R becomes a left R⟨∂,
+�
+, E⟩-module and multiplication
+in R⟨∂,
+�
+, E⟩ corresponds to composition of additive maps on R.
+Since elements of C always commute with ∂,
+�
+, E in R⟨∂,
+�
+, E⟩, it follows
+that R⟨∂,
+�
+, E⟩ is a C-algebra whenever R is commutative. For computing in
+the ring R⟨∂, � , E⟩ of IDO, we use the identities (11)–(16) as rewrite rules in
+the following way. If the left hand side of one of these identities appears in an
+expression of an operator, we replace it by the right hand side to obtain a new
+expression for the same operator.
+If rewrite rules can be applied to a given expression in different ways, then
+it may happen that useful consequences of the defining relations (11)–(16) are
+discovered. A simple instance starts with the expression
+�
+· ∂ ·
+�
+, to which we
+can apply either (12) or (13) to obtain the expressions
+�
+and
+�
+− E ·
+�
+for the
+same operator. Hence, by taking their difference, we see that the identity
+E ·
+�
+= 0
+(17)
+holds in R⟨∂,
+�
+, E⟩. Moreover, for every f ∈ R, the expression
+�
+· ∂ · f can be
+rewritten by (11) and by (13). Thereby we obtain the expressions
+�
+·f ·∂+
+�
+·∂f
+and f − E · f for the same operator, which implies the identity
+�
+· f · ∂ = f − E · f −
+�
+· ∂f.
+(18)
+By letting both sides of this identity in R⟨∂,
+�
+, E⟩ act on any g ∈ R, we show
+that integration by parts
+�
+f∂g = fg − Efg −
+�
+(∂f)g
+holds in R. Furthermore, by considering also the newly obtained identity (18)
+as rewrite rule, for every f ∈ R, we can rewrite the expression
+�
+·f ·∂ ·
+�
+by (12)
+and by (18). Substituting f by � f in the difference f ·� −E·f ·� −� ·∂f ·� −� ·f
+of the results, we obtain the identity
+�
+· f ·
+�
+=
+�
+f ·
+�
+−
+�
+·
+�
+f − E ·
+�
+f ·
+�
+.
+(19)
+By acting with both sides of this identity on any g ∈ R, we obtain an alternative
+proof for the Rota-Baxter identity with evaluation (8). Note that, for f = 1, we
+also obtain the following identities from (14)–(16) and (19).
+∂ · E = 0,
+�
+· E =
+�
+1 · E,
+E · E = E,
+(20)
+�
+·
+�
+=
+�
+1 ·
+�
+−
+�
+·
+�
+1 − E ·
+�
+1 ·
+�
+(21)
+In Table 1, we collect the identities (11)–(21) as a rewrite system for expres-
+sions of operators in the ring of IDO. In fact, we drop (14) since it is redundant
+in the presence of (11) and (20).
+16
+
+Theorem 4.2. Let (R, ∂,
+�
+) be an integro-differential ring. Then, by repeatedly
+applying the rewrite rules of Table 1 in any order, every element of the ring
+R⟨∂,
+�
+, E⟩ can be written as a sum of expressions of the form
+f · ∂j,
+f ·
+�
+· g,
+f · E · g · ∂j,
+or
+f · E · h ·
+�
+· g
+where j ∈ N0, f, g ∈ R, and h ∈
+�
+R.
+Note that the expressions specified in the above theorem are irreducible in
+the sense that they cannot be rewritten any further by any rewrite rules from
+Table 1. The above derivation of identities (17)–(21) is similar to Knuth-Bendix
+completion [19] and Buchberger’s algorithm for computing Gröbner bases [4, 22].
+So, one can show that Table 1 represents all consequences of Definition 4.1
+in the sense that every identity in R⟨∂,
+�
+, E⟩ can be proven by applying the
+rewrite rules in the table and by exploiting identities in R.
+Moreover, the
+irreducible forms of operators specified in the above theorem are unique up to
+multiadditivity and commutativity.
+In the appendix, we will give a precise statement (Theorem A.3) of this
+by giving an explicit construction of the ring R��∂,
+�
+, E⟩ as a quotient of an
+appropriate tensor ring. Then, the translation of Table 1 into a tensor reduction
+system facilitates the proof. In fact, this proof is carried out in Theorem A.5 for
+a more general class of operators including linear functionals, which are useful
+for dealing with boundary problems, for instance.
+Remark 4.3. In the literature, integro-differential operators were considered
+only with multiplicative evaluation so far. Integro-differential operators were
+first introduced in [29, 30] over a field of constants using a parametrized Gröb-
+ner basis in infinitely many variables and a basis of the commutative coefficient
+algebra. Integro-differential operators with polynomial coefficients over a field
+of characteristic zero were also studied using generalized Weyl algebras [2], skew
+polynomials [28], and noncommutative Gröbner bases [26]. A general construc-
+tion of rings of linear operators over commutative operated algebras is presented
+in [9]. In particular, integro-differential operators are discussed in that setting
+and also differential Rota-Baxter operators are investigated. Tensor reduction
+systems have already been used in [14, 13] for the construction of IDO includ-
+ing functionals in the special case that the evaluation and all functionals are
+multiplicative.
+There, also additional operators arising from linear substitu-
+tions were included.
+In particular, these cover integro-differential-time-delay
+operators, which were already constructed algebraically in [25], see also [5].
+∂ · f = f · ∂ + ∂f
+�
+· f · ∂ = f − E · f −
+�
+· ∂f
+∂ · E = 0
+� · f · E = � f · E
+∂ ·
+�
+= 1
+�
+· f ·
+�
+=
+�
+f ·
+�
+−
+�
+·
+�
+f − E ·
+�
+f ·
+�
+E · f · E = Ef · E
+�
+· ∂ = 1 − E
+E · E = E
+�
+· E =
+�
+1 · E
+E ·
+�
+= 0
+�
+·
+�
+=
+�
+1 ·
+�
+−
+�
+·
+�
+1 − E ·
+�
+1 ·
+�
+Table 1: Rewrite rules for operator expressions
+17
+
+To refer to integro-differential operators of special form, we use the following
+notions.
+Definition 4.4. Let L ∈ R⟨∂,
+�
+, E⟩, then we call L
+1. a differential operator, if there are f0, . . . , fn ∈ R such that
+L =
+n
+�
+i=0
+fi · ∂i,
+where we call L monic, if fn = 1,
+2. an integral operator, if there are f1, . . . , fn, g1, . . . , gn ∈ R such that
+L =
+n
+�
+i=1
+fi ·
+�
+· gi,
+3. an initial operator, if there are f1, . . . , fn ∈ R and differential and integral
+operators L1, . . . , Ln such that
+L =
+n
+�
+i=1
+fi · E · Li.
+In particular, we call an initial operator L monic, if there is a differential
+operator L1 and an integral operator L2 such that
+L = E · (L1 + L2).
+In R⟨∂, � , E⟩, using Table 1, one can check that the differential operators
+are the elements of the subring generated by R and ∂, the integral operators
+are the elements of the R-bimodule generated by
+�
+, the initial operators are the
+elements of the two-sided ideal generated by E, and the monic initial operators
+are the elements of the right ideal generated by E. Theorem 4.2 says that every
+integro-differential operator can be written as the sum of a differential operator,
+an integral operator, and an initial operator. In fact, by the stronger results in
+the appendix, this decomposition of integro-differential operators even is unique.
+Remark 4.5. We outline how the construction of integro-differential opera-
+tors changes when the evaluation is multiplicative, see also [31, 14, 13]. For
+multiplicative evaluation, we have Efg = (Ef)Eg for all f, g ∈ R. So, for the
+algebraic construction of corresponding operators as in Definition 4.1, we need
+to impose in addition that
+E · f = Ef · E
+(22)
+for all f ∈ R. This does not give rise to new consequences other than (17)–(21),
+but together with (20) it makes (16) redundant. Hence, in Table 1, we can
+replace (16) by (22). Moreover, (22) allows to reduce the evaluation term in
+(18) and, since E
+�
+f = 0 for all f ∈ R, to omit the evaluation terms in (19) and
+18
+
+(21). Consequently, the irreducible forms of operators given in Theorem 4.2 can
+be simplified to
+f · ∂j,
+f ·
+�
+· g,
+f · E · ∂j,
+where j ∈ N0 and f, g ∈ R. Evidently, this ring of operators is isomorphic to
+R⟨∂,
+�
+, E⟩ factored by the two-sided ideal (E · f − Ef · E | f ∈ R).
+4.1
+IDO over integral domains
+In many concrete situations, when computing with differential operators or dif-
+ferential equations with scalar coefficients, the order of a product of differential
+operators is the sum of the orders of the factors.
+This is equivalent to the
+ring R of coefficients being an integral domain, i.e. a commutative ring without
+nontrivial zero divisors. For the rest of this section, we only consider integro-
+differential rings that are integral domains. Under this assumption, we can in-
+vestigate further properties of computations in the ring of IDO. For instance, an
+integro-differential equation can be reduced to a differential equation by differ-
+entiation, see Lemma 4.8 below. Additionally, investigating the action of IDOs
+on integro-differential rings algebraically leads to analyzing two-sided ideals.
+As explained earlier, elements of R⟨∂,
+�
+, E⟩ naturally act as additive maps
+on R. Likewise, they act naturally on any integro-differential ring extension of
+R. In Theorem 4.11, we also provide conditions when this action is faithful, i.e.
+0 ∈ R⟨∂,
+�
+, E⟩ is the only element that induces the zero map.
+Let (S,
+�
+, ∂) be the integro-differential ring defined in Example 2.13 and as-
+sume C is an integral domain. Then, S⟨∂, � , E⟩ does not act faithfully on S, since
+E · ln(x) acts like zero. However, with the integro-differential subring R := C[x]
+of S, R⟨∂,
+�
+, E⟩ acts faithfully on S by Theorem 4.11 below. To see this, let
+L := �n
+i=0 fi ·∂i ∈ R⟨∂,
+�
+, E⟩ and let k ∈ N such that coeff(fn, xk) ̸= 0. Apply-
+ing E·L to xn−k ln(x)n ∈ S, we obtain (E·L)xn−k ln(x)n = n! coeff(fn, xk) ̸= 0.
+As a preparatory step, we need some basic statements involving multiplica-
+tion with constants that are valid for integro-differential rings that are integral
+domains. Linear independence over constants is tied to the Wronskian. Follow-
+ing the analytic definition, the Wronskian of elements f1, . . . , fn of a commuta-
+tive differential ring is defined by
+W(f1, . . . , fn) := det
+��
+∂i−1fj
+�
+i,j=1,...,n
+�
+.
+Lemma 4.6. Let (R, ∂) be a differential ring that is an integral domain with
+ring of constants C and let f1, . . . , fn ∈ R, n ≥ 1.
+1. f1, . . . , fn are linearly independent over C if and only if their Wronskian
+W(f1, . . . , fn) is zero.
+2. If f1, . . . , fn are linearly independent over C, then g1, . . . , gn−1 are linearly
+independent over C, where gi := W(fi, fn).
+Proof. For showing the first statement, we note that neither linear independence
+nor zeroness of the Wronskian changes, if we replace R and hence C by their
+19
+
+quotient fields. For differential fields, a proof can be found in [15], which implies
+the statement given here.
+For showing the second statement, we assume that f1, . . . , fn are linearly
+independent over C and we let c1, . . . , cn−1 ∈ C such that �n−1
+i=1 cigi = 0.
+By definition of gi and multilinearity of the Wronskian over C, we conclude
+W(�n−1
+i=1 cifi, fn) = 0. By assumption on f1, . . . , fn, this implies �n−1
+i=1 cifi = 0
+and hence c1 = . . . = cn−1 = 0.
+Lemma 4.7. Let (R, ∂,
+�
+) be an integro-differential ring that is an integral
+domain and let C be its ring of constants. Then, elements of C commute with
+all elements of R⟨∂,
+�
+, E⟩. Moreover, for any L ∈ R⟨∂,
+�
+, E⟩ and any nonzero
+c ∈ C, we have that L = 0 if and only if c · L = 0.
+Proof. By construction, constants commute with ∂,
+�
+, E ∈ R⟨∂,
+�
+, E⟩. Since
+R is commutative, it follows that elements of C commute with all elements
+of R⟨∂,
+�
+, E⟩. Therefore, R⟨∂,
+�
+, E⟩ is a unital C-algebra and hence Q(C) ⊗C
+R⟨∂,
+�
+, E⟩ is a unital C-algebra as well, with multiplication (c1⊗L1)·(c2⊗L2) =
+(c1c2) ⊗ (L1 · L2). Now, 1 ⊗ L = c−1 ⊗ (c · L) implies L = 0 if c · L = 0.
+Now, we are ready to have a closer look at certain computations with IDO.
+The following two lemmas construct left or right multiples of IDO by differential
+operators such that the product does not involve the integration operator any-
+more. The tricky part will be to ensure that the product is a nonzero operator
+again.
+Lemma 4.8. Let (R, ∂,
+�
+) be an integro-differential ring that is an integral
+domain.
+Let C be the ring of constants of R and let L ∈ R⟨∂, � , E⟩ be not
+an initial operator.
+Then, there exist nonzero h1, . . . , hn ∈ R such that the
+following product is a nonzero differential operator.
+(h1 · ∂ − ∂h1) · . . . · (hn · ∂ − ∂hn) · L
+Proof. There are n ∈ N, a nonzero c ∈ C, differential operators L0, . . . , Ln ∈
+R⟨∂,
+�
+, E⟩, and f1, . . . , fn, g1, . . . , gn ∈ R such that f1, . . . , fn are linearly inde-
+pendent over C and c · L = L0 + �n
+i=1 fi ·
+��
+· gi + E · Li
+�
+. Since L is not an
+initial operator, L0 and g1, . . . , gn are not all zero by Lemma 4.7. If L0 = 0,
+then we assume without loss of generality that g1 ̸= 0.
+To remove the sum �n
+i=1 fi·
+��
+· gi + E · Li
+�
+, we will multiply c·L iteratively
+by n first-order differential operators from the left as follows. If n > 0, then by
+(11) and (12) we have (fn ·∂ − ∂fn)·fi ·
+�
+·gi = fnfigi + (fn∂fi − (∂fn)fi)·
+�
+·gi
+for all i. Therefore, using (14), we obtain
+(fn · ∂ − ∂fn) · c · L = (fn · ∂ − ∂fn) · L0 +
+n
+�
+i=1
+fnfigi
++
+n−1
+�
+i=1
+(fn∂fi − (∂fn)fi) ·
+��
+· gi + E · Li
+�
+,
+20
+
+which has the form ˜L0+�n−1
+i=1 ˜fi ·
+��
+· gi + E · Li
+�
+similar to c·L. Note that also
+the following two properties of the above representation of c · L are preserved.
+First, if L0 is nonzero, the differential operator ˜L0 := (fn · ∂ − ∂fn) · L0 +
+�n
+i=1 fnfigi is nonzero too, since fn ̸= 0. Second, ˜fi := fn∂fi − (∂fn)fi, for
+i ∈ {1, . . . , n − 1}, are again linearly independent over C by Lemma 4.6. Now,
+we let hn := fnc such that (hn · ∂ − ∂hn) · L = (fn · ∂ − ∂fn) · c · L.
+If n = 1, we only need to show that (hn · ∂ − ∂hn) · L = ˜L0 is nonzero. As
+remarked above, if L0 ̸= 0 then ˜L0 ̸= 0. On the other hand, if L0 = 0, then
+˜L0 = f 2
+1 g1, which is nonzero as well due to the assumptions.
+If n > 1, we continue by repeating what we did with c · L above. That is,
+noting ˜f1, . . . , ˜fn−1 are linearly independent over C, we multiply (hn · ∂ − ∂hn) ·
+L = ˜L0 + �n−1
+i=1 ˜fi ·
+��
+· gi + E · Li
+�
+from the left by hn−1 · ∂ − ∂hn−1, where
+hn−1 := ˜fn−1. We iterate this a total of n − 1 times, the result is a differential
+operator. To see that it is also nonzero, we focus on the last step, i.e. we refer
+to the case n = 1 above.
+An immediate consequence of the previous lemma is that a left ideal in
+R⟨∂,
+�
+, E⟩ is nontrivial if and only if it contains a nonzero differential operator
+or a nonzero initial operator. Left ideals in R⟨∂,
+�
+, E⟩ arise, for instance, as sets
+of operators annihilating a given subset of a left R⟨∂, � , E⟩-module. As another
+consequence, for two-sided ideals, we obtain Lemma 4.10 below.
+Lemma 4.9. Let (R, ∂,
+�
+) be an integro-differential ring that is an integral
+domain. Let C be the ring of constants of R and let L ∈ R⟨∂,
+�
+, E⟩ be a nonzero
+monic initial operator. Then, there exist nonzero h1, . . . , hn ∈ R and a nonzero
+differential operator ˜L ∈ R⟨∂,
+�
+, E⟩ such that
+L · (hn · ∂ + 2∂hn) · . . . · (h1 · ∂ + 2∂h1) = E · ˜L.
+Proof. There are n ∈ N, a nonzero c ∈ C, a differential operator L0 ∈ R⟨∂,
+�
+, E⟩,
+nonzero f1, . . . , fn ∈ � R, and nonzero g1, . . . , gn ∈ R such that g1, . . . , gn are
+linearly independent over C and L·c = E·L0+�n
+i=1 E·fi·
+�
+·gi. By Lemma 4.7,
+L · c = c · L is nonzero.
+To remove the sum �n
+i=1 E · fi ·
+�
+· gi, we will multiply L · c iteratively by
+n first-order differential operators from the right as follows. If n > 0, then we
+have
+E · fi ·
+�
+· gign · ∂ = E · figign − Efi · E · gign − E · fi ·
+�
+· ((∂gi)gn + gi∂gn)
+for all i by (18) and by (16). Therefore, by Efi = 0, we obtain
+L · c · (gn · ∂ + 2∂gn) = E · L0 · (gn · ∂ + 2∂gn) +
+n
+�
+i=1
+E · fi ·
+�
+· (gign · ∂ + 2gi∂gn)
+= E · L1 +
+n−1
+�
+i=1
+E · fi ·
+�
+· (gi∂gn − (∂gi)gn),
+21
+
+with L1 := L0 · (gn · ∂ + 2∂gn) + �n
+i=1 figign.
+Note that gi∂gn − (∂gi)gn,
+for i ∈ {1, . . . , n − 1}, are again linearly independent over C by Lemma 4.6.
+Consequently, with hn := cgn being nonzero, the operator L · (hn · ∂ + 2∂hn) is
+expressed like L · c above, just with L0 replaced by L1, with different gi, and
+without the last summand.
+By iterating this process of multiplying by a differential operator that is
+chosen as above, we obtain a right multiple of L that is of the form E · Ln with
+a differential operator Ln. It remains to show that Ln is nonzero. After the
+last step, i.e. passing from E · Ln−1 + E · f1 ·
+�
+· ˜g1, with differential operator
+Ln−1 and nonzero ˜g1 ∈ R, to E · Ln, we have Ln = Ln−1 · (˜g1 · ∂ + 2∂˜g1) +
+f1˜g2
+1. If the differential operator Ln−1 is nonzero, then also Ln is nonzero, since
+˜g1 ̸= 0. Otherwise, we have Ln = f1˜g2
+1, which is nonzero as well due to the
+assumptions.
+Lemma 4.10. Let (R, ∂,
+�
+) be an integro-differential ring that is an integral
+domain and let I ⊆ R⟨∂,
+�
+, E⟩ be a two-sided ideal. If I contains an operator
+that is not an initial operator, then I contains an operator of the form f ·E with
+nonzero f ∈ R.
+Proof. By Lemma 4.8, I contains a nonzero differential operator L = �n
+i=0 fi·∂i.
+Let k ∈ {0, . . ., n} be minimal such that fk ̸= 0. Then, fk · E ∈ I since
+L ·
+� k · E =
+n
+�
+i=k
+fi · ∂i ·
+� k · E =
+n
+�
+i=k
+fi · ∂i−k · E = fk · E.
+Based on the above lemmas, we can find conditions when the action of
+R⟨∂,
+�
+, E⟩ on an integro-differential ring is faithful. In this case, the annihilator
+AnnR⟨∂,�
+,E⟩(S) = {L ∈ R⟨∂,
+�
+, E⟩ | ∀f ∈ S : Lf = 0}
+is a two-sided ideal in R⟨∂,
+�
+, E⟩. In particular, we show that the annihilator
+only contains initial operators and, if C is a field, can be generated by monic
+initial operators.
+Theorem 4.11. Let (S, ∂,
+�
+) be an integro-differential ring with its ring of
+constants C being an integral domain. Let R be an integro-differential subring of
+S that is an integral domain and contains C. Then, R⟨∂,
+�
+, E⟩ acts faithfully on
+S if and only if there is no nonzero differential operator L ∈ R⟨∂,
+�
+, E⟩ such that
+E · L vanishes on all of S. In any case, the two-sided ideal I = AnnR⟨∂,�
+,E⟩(S)
+only contains initial operators. Moreover, if C is a field, I has a generating set
+consisting only of monic initial operators.
+Proof. Since operators of the form f · E, with nonzero f ∈ R, do not vanish on
+1 ∈ S, the ideal I only contains initial operators by Lemma 4.10. If there is a
+nonzero differential operator L ∈ R⟨∂,
+�
+, E⟩ such that E · L vanishes on all of
+S, then R⟨∂,
+�
+, E⟩ obviously does not act faithfully on S, since E · L is nonzero
+as well.
+22
+
+Now, we assume that there is no nonzero differential operator L such that
+E · L ∈ I.
+Let L ∈ I, then there are nonzero c ∈ C, f1, . . . , fn ∈ R, and
+nonzero monic initial operators L1, . . . , Ln ∈ R⟨∂,
+�
+, E⟩ such that f1, . . . , fn are
+C-linearly independent and c · L = �n
+i=1 fi · Li. If we would have n ̸= 0, then
+we would conclude L1, . . . , Ln ∈ I, since �n
+i=1 fiLig = cLg = 0 and Lig ∈ C
+for all g ∈ S. Therefore, by Lemma 4.9, there would be a nonzero differential
+operator ˜L ∈ R⟨∂,
+�
+, E⟩ such that E · ˜L ∈ I in contradiction to the assumption.
+Hence, n = 0, which implies c · L = 0. By Lemma 4.7, it follows that L = 0,
+which shows that R⟨∂,
+�
+, E⟩ acts faithfully on S.
+Finally, if C is a field, we show that I has a generating set consisting only
+of monic initial operators. Let L ∈ I, then L is an initial operator. Now, there
+are nonzero f1, . . . , fn ∈ R, and nonzero monic initial operators L1, . . . , Ln ∈
+R⟨∂,
+�
+, E⟩ such that f1, . . . , fn are C-linearly independent and L = �n
+i=1 fi · Li.
+For all g ∈ S, we have Lig ∈ C and �n
+i=1 fiLig = Lg = 0, which implies Lig = 0
+for all i. Therefore, L1, . . . , Ln ∈ I, which shows that I is generated by nonzero
+monic initial operators.
+Assuming additional properties of R resp. of the action of R⟨∂, � , E⟩, gen-
+erating sets of annihilators can be narrowed down even more. In particular,
+the following two corollaries consider the situation when R is a field or E is
+multiplicative.
+Corollary 4.12. Let (S, ∂, � ) be an integro-differential ring with its ring of
+constants C being a field. Let R be an integro-differential subring of S that is a
+field and contains C. Then, the two-sided ideal AnnR⟨∂,
+�
+,E⟩(S) has a generating
+set consisting only of operators of the form E · �n
+i=0 fi · ∂i, with f0, . . . , fn ∈ R
+where f0 ∈
+�
+R is nonzero and fn = 1.
+Proof. By Theorem 4.11, I := AnnR⟨∂,
+�
+,E⟩(S) has a generating set consisting
+only of monic initial operators. Now, let L ∈ I be a monic initial operator and
+let A be the set of all elements in I that have the form specified in the statement
+of the corollary. By Lemma 4.9, we obtain nonzero h1, . . . , hm ∈ R such that
+L · M = E · ˜L with M := (hm · ∂ + 2∂hm) · . . . · (h1 · ∂ + 2∂h1) ∈ R⟨∂,
+�
+, E⟩
+and some nonzero differential operator ˜L = �n
+i=0 ˜fi · ∂i ∈ R⟨∂,
+�
+, E⟩ with
+˜fn ̸= 0. Then, by repeated use of (11), there are f0, . . . , fn ∈ R with fn = 1
+such that ˜L · ˜f −1
+n
+= �n
+i=0 fi · ∂i. Let k be minimal such that fk ̸= 0, then
+˜L · ˜f −1
+n
+· � k = �n−k
+i=0 fi+k · ∂i. Since E · ˜L · ˜f −1
+n
+· � k ∈ I vanishes on 1 ∈ S,
+we conclude fk ∈
+�
+R and hence E · �n−k
+i=0 fi+k · ∂i ∈ A. Since R is a field, we
+can verify by (11) and (12) that h−1
+i
+·
+�
+· hi ∈ R⟨∂,
+�
+, E⟩ is a right inverse of
+hi·∂+2∂hi for every i. Hence, there exists ˜
+M ∈ R⟨∂,
+�
+, E⟩ such that M · ˜
+M = 1.
+Then, E ·
+��n−k
+i=0 fi+k · ∂i�
+· ∂k · ˜fn · ˜
+M = E · ˜L · ˜
+M = L · M · ˜
+M = L, i.e. L is in
+the ideal generated by A. Altogether, this shows that I is generated by A.
+Whenever E is multiplicative on R, the action of R⟨∂,
+�
+, E⟩ cannot be faithful
+on R since the operator E · f acts as zero map for any f ∈ R with Ef = 0.
+23
+
+By Lemma 2.3, Ef = 0 is equivalent to f ∈
+�
+R and one can always choose
+f =
+�
+1, for example. In Remark 4.5, we already discussed factoring the ring of
+operators by the relations (22) immediately arising from multiplicativity of E.
+The following corollary shows that the resulting quotient always acts faithfully.
+Corollary 4.13. Let (S, ∂,
+�
+) be an integro-differential ring with its ring of
+constants C being an integral domain. Let R be an integro-differential subring
+of S that is an integral domain and contains C. If Efg = (Ef)Eg for all f ∈ R
+and g ∈ S, then the two-sided ideal AnnR⟨∂,
+�
+,E⟩(S) is generated by the set
+{E · f − Ef · E | f ∈ R}.
+Proof. Let I := AnnR⟨∂,�
+,E⟩(S) and let J ⊆ R⟨∂,
+�
+, E⟩ be the two-sided ideal
+generated by the set {E · f − Ef · E | f ∈ R}. By assumption on E, we have
+that J ⊆ I, so it only remains to show I ⊆ J.
+By Theorem 4.11, every L ∈ I is an initial operator. Following the form
+of initial operators in Remark 4.5, there are f0, . . . , fn ∈ R such that L =
+�n
+i=0 fi ·E·∂i modulo J. By J ⊆ I, we have that �n
+i=0 fi ·E·∂i ∈ I. Therefore,
+�n
+i=0 fi·E·∂i vanishes on
+� k1 ∈ S for all k ∈ {0, . . . , n}. This implies inductively
+that f0, . . . , fn are all zero. Consequently, we have L ∈ J.
+5
+Equational prover in calculus
+In this section, we illustrate how results from analysis can be proven via compu-
+tations in the ring of integro-differential operators. The general approach con-
+sists in formulating an analytic statement as an identity of integro-differential
+operators and then prove this identity algebraically in R⟨∂,
+�
+, E⟩, with minimal
+assumptions on the integro-differential ring (R, ∂,
+�
+) used in the coefficients.
+Note that we prove results directly by a computation with integro-differential
+operators, instead of doing the whole computation with elements of R or any
+other left R⟨∂,
+�
+, E⟩-module.
+An identity in R⟨∂,
+�
+, E⟩ can be proven by comparing irreducible forms
+of the left hand side and right hand side. Recall that such irreducible forms
+are given by Theorem 4.2 and can be computed systematically by the rewrite
+rules in Table 1. In this sense, the rewrite rules provide an equational prover
+for integro-differential operators provided one can decide equality of irreducible
+forms in R⟨∂,
+�
+, E⟩, which includes deciding identities in R. In practice, this is
+often possible for concrete irreducible forms.
+Once an identity L1 = L2 is proven in R⟨∂,
+�
+, E⟩, we immediately infer the
+corresponding identity in R, i.e. L1f = L2f for all f ∈ R, by the canonical
+action of R⟨∂,
+�
+, E⟩ on R. Moreover, by acting on any other left R⟨∂,
+�
+, E⟩-
+module, we obtain the analogous identity also in those modules. Furthermore,
+any concrete computation with operators acting on functions uses only finitely
+many derivatives. Consequently, by inspecting every step of the computation,
+an identity proven for infinitely differentiable functions can also be proven for
+functions that are only sufficiently often differentiable.
+24
+
+5.1
+Variation of constants for scalar equations
+In this section, we deal with the method of variation of constants for comput-
+ing solutions of inhomogeneous ODEs in terms of integro-differential operators.
+First, we recall the analytic statement, see e.g. Theorem 6.4 in Chapter 3 of [6].
+Consider the inhomogeneous linear ODE
+y(n)(x) + an−1(x)y(n−1)(x) + · · · + a0(x)y(x) = f(x).
+Assume that z1(x), . . . , zn(x) is a fundamental system of the homogeneous equa-
+tion, i.e. z(n)
+i
+(x) + an−1(x)z(n−1)
+i
+(x) + · · · + a0(x)zi(x) = 0 and the Wronskian
+w(x) := W(z1(x), . . . , zn(x)) is nonzero. Then,
+z∗(x) :=
+n
+�
+i=1
+(−1)n−izi(x)
+� x
+x0
+W(z1(t), . . . , zi−1(t), zi+1(t), . . . , zn(t))
+w(t)
+f(t) dt
+(23)
+is a particular solution of the inhomogeneous equation above.
+Algebraically, we model scalar functions by fixing a commutative integro-
+differential ring (R, ∂,
+�
+) and, as indicated above, we model computations with
+scalar equations by computations in the corresponding ring of integro-differential
+operators R⟨∂,
+�
+, E⟩. From the operator viewpoint, mapping the inhomogeneous
+part f to a solution of the equation Ly = f amounts to constructing a right
+inverse of the differential operator L.
+In general, for an operator L and a
+right inverse H, a particular solution of the inhomogeneous equation is given by
+z∗ = Hf, since
+Lz∗ = L(Hf) = (L · H)f = 1f = f.
+Recall that
+�
+is a right inverse of ∂ by definition.
+Now, we consider an
+arbitrary monic first-order differential operator
+L = ∂ + a,
+with a ∈ R, and assume that z ∈ R is a solution of Ly = 0, i.e. ∂z + az = 0.
+Then, using (11), we compute
+(∂ + a) · z = ∂ · z + a · z = z · ∂ + ∂z + az = z · ∂.
+If, moreover, we assume that z has a multiplicative inverse z−1 ∈ R, we obtain
+(∂ + a) · (z ·
+�
+· z−1) = z · ∂ ·
+�
+· z−1 = z · z−1 = 1.
+Hence
+H = z ·
+�
+· z−1
+is a right inverse of L in R⟨∂,
+�
+, E⟩.
+We also outline the computation for a second order differential operator
+L = ∂2 + a1 · ∂ + a0,
+25
+
+with a1, a0 ∈ R. If z ∈ R is a solution of Ly = 0, then
+L · z = z · ∂2 + (2∂z + a1z) · ∂.
+Assume that there exist two solutions z1, z2 ∈ R of Ly = 0 such that their
+Wronskian
+w = z1∂z2 − z2∂z1
+has a multiplicative inverse 1
+w ∈ R. Let
+H = −z1 ·
+�
+· z2
+w + z2 ·
+�
+· z1
+w .
+In the ring of operators, we can compute
+∂ · zi
+w = zi
+w · ∂ + ∂zi
+w − zi∂w
+w2 .
+Then, using the normal forms of L · zi and ∂ · zi
+w , we obtain
+L · H = −z1 · ∂ · z2
+w − 2(∂z1) z2
+w − a1z1 z2
+w + z2 · ∂ · z1
+w + 2(∂z2) z1
+w + a1z2 z1
+w
+= −z1 · ∂ · z2
+w + z2 · ∂ · z1
+w + 2 w
+w = −z1∂ z2
+w + z2∂ z1
+w + 2 = 1.
+Hence H is a right inverse of L in R⟨∂,
+�
+, E⟩.
+More generally, we have the following formulation of the method of variation
+of constants for integro-differential operators.
+Recall from Section 4.1, that
+the Wronskian W(f1, . . . , fn) of elements f1, . . . , fn ∈ R is defined completely
+analogous to the analytic situation.
+Theorem 5.1. Let (R, ∂,
+�
+) be a commutative integro-differential ring and let
+L = ∂n + �n−1
+i=0 ai · ∂i ∈ R⟨∂,
+�
+, E⟩, n ≥ 1, with a0, . . . , an−1 ∈ R. Assume that
+z1, . . . , zn ∈ R are such that Lzi = 0 and w := W(z1, . . . , zn) has a multiplicative
+inverse
+1
+w ∈ R. Then, with
+H :=
+n
+�
+i=1
+(−1)n−izi ·
+�
+· W(z1, . . . , zi−1, zi+1, . . . , zn)
+w
+we have that L · H = 1 in R⟨∂,
+�
+, E⟩.
+Proof. The cases n = 1 and n = 2 have been shown above. In principle, an
+analogous computation could be done for any concrete n ≥ 3 as well. To obtain
+a finite proof for all n ≥ 3 at once, we will utilize a more general framework in
+Section 5.3.
+5.2
+Linear systems and operators with matrix coefficients
+Algebraically, computing with linear systems of differential equations can be
+modelled by integro-differential operators over some noncommutative integro-
+differential ring, whose elements correspond to matrices. Such noncommutative
+integro-differential rings can be obtained from any scalar integro-differential
+26
+
+ring, since one can equip the ring of n × n matrices with a derivation and an
+integration defined entrywise, see Lemma 2.7.
+Recall that Definition 4.1 and Theorem 4.2 hold also for operators with
+coefficients in noncommutative integro-differential rings, in particular for oper-
+ators with matrix coefficients. Consequently, any computation using a general
+noncommutative integro-differential ring is automatically valid for concrete ma-
+trices of any size, without the need for entrywise computations. This allows for
+compact proofs for matrices of general size. For switching to entrywise com-
+putations, one can view operators with matrix coefficients equivalently also as
+matrices of operators with scalar coefficients by identifying operators ∂,
+�
+, E
+with corresponding diagonal matrices of operators. The following lemma shows
+that also computations are equivalent in both viewpoints. Here, we use the
+notation Ei,j(L) := (δi,kδj,lL)k,l=1,...,n for matrices with only one nonzero entry
+L ∈ R⟨∂,
+�
+, E⟩.
+Lemma 5.2. Let (R, ∂,
+�
+) an integro-differential ring and let n ≥ 1. Then,
+there is exactly one ring homomorphism
+ϕ : Rn×n⟨∂,
+�
+, E⟩ → R⟨∂,
+�
+, E⟩n×n
+with ϕ(A) = A for A ∈ Rn×n and ϕ(L) = diag(L, . . . , L) for L ∈ {∂,
+�
+, E}.
+This ϕ is an isomorphism and its inverse homomorphism
+ψ : R⟨∂, � , E⟩n×n → Rn×n⟨∂, � , E⟩
+can be given by ψ(Ei,j(f)) = Ei,j(f) and ψ(Ei,j(L)) = Ei,j(1) · L for all f ∈ R
+resp. L ∈ {∂,
+�
+, E}.
+Proof. First, we check that the definition of ϕ indeed provides a unique and
+well-defined homomorphism of rings. Uniqueness follows from the fact that ϕ is
+defined on a generating set of Rn×n⟨∂,
+�
+, E⟩. For proving well-definedness, we
+need to verify that the definition of ϕ respects all identities of generators given
+in Definition 4.1 as follows. Evidently, we have ϕ(1) = In. It is immediate to see
+that ϕ(∂) · ϕ(
+�
+) = ϕ(1) − ϕ(E) and ϕ(
+�
+) · ϕ(∂) = ϕ(1) hold. For L ∈ {∂,
+�
+, E},
+we verify in R⟨∂,
+�
+, E⟩n×n that ϕ(L) commutes with all elements of Cn×n and
+that, for all A ∈ Rn×n, we have ϕ(L) · ϕ(A) · ϕ(E) = ϕ(LA) · ϕ(E).
+More
+explicitly, using (14)–(16) in R⟨∂,
+�
+, E⟩, we compute
+ϕ(L) · ϕ(A) · ϕ(E) = diag(L, . . . , L) · A · diag(E, . . . , E)
+= (L · ai,j · E)i,j=1,...,n = (Lai,j · E)i,j=1,...,n = ϕ(LA) · ϕ(E).
+Analogously, using (11) in R⟨∂,
+�
+, E⟩, we can also verify that ϕ(∂) · ϕ(A) =
+ϕ(A) · ϕ(∂) + ϕ(∂A) for all A ∈ Rn×n.
+Similarly, we need to verify that the definition of ψ on a generating set of
+R⟨∂,
+�
+, E⟩n×n gives rise to a well-defined homomorphism. For example, using
+27
+
+(14)–(16) in Rn×n⟨∂,
+�
+, E⟩, we compute
+ψ(Ei,j(L)) · ψ(Ep,q(f)) · ψ(Ek,l(E)) = Ei,j(1) · L · Ep,q(f) · Ek,l(1) · E
+= Ei,j(1) · L · δq,kEp,l(f) · E = Ei,j(1) · δq,kLEp,l(f) · E = δj,pδq,kEi,l(Lf) · E
+= Ei,q(δj,pLf) · Ek,l(1) · E = ψ(Ei,q(δj,pLf)) · ψ(Ek,l(E))
+for all f ∈ R, L ∈ {∂,
+�
+, E}, and all i, j, k, l, p, q ∈ {1, . . ., n}.
+Finally, ψ is the inverse of ϕ, since we have ψ(ϕ(A)) = A, ϕ(ψ(Ei,j(f))) =
+Ei,j(f), ψ(ϕ(L)) = L, and ϕ(ψ(Ei,j(L))) = Ei,j(L) for the generators.
+Integro-differential operators with coefficients from Rn×n constructed this
+way have natural actions on Rn×n and on Rn.
+By the above lemma, for
+any left R⟨∂,
+�
+, E���-module M, there is a natural way of viewing M n as a
+left Rn×n⟨∂, � , E⟩-module.
+It is straightforward to show that the action of
+Rn×n⟨∂,
+�
+, E⟩ on M n is faithful if and only if R⟨∂,
+�
+, E⟩ acts faithfully on M.
+5.3
+Variation of constants for first-order systems
+Instead of higher order scalar ODEs, we now consider variation of constants for
+first-order systems, see Theorem 3.1 in Chapter 3 of [6] for example. Analyti-
+cally, it can be stated as follows. For an n × n matrix A(x) and a vector f(x)
+of size n, we consider the first-order system given by
+y′(x) + A(x)y(x) = f(x).
+If Φ(x) is a fundamental matrix of the homogeneous system, i.e. it satisfies
+Φ′(x) + A(x)Φ(x) = 0 and det(Φ(x)) ̸= 0, then a particular solution of the
+inhomogeneous system is given by
+z∗(x) = Φ(x)
+� x
+x0
+Φ(t)−1f(t) dt.
+Theorem 5.3. Let (R, ∂,
+�
+) be an integro-differential ring. Assume that a ∈ R
+is such that there exists z ∈ R that satisfies ∂z +az = 0 and has a multiplicative
+right inverse z−1 ∈ R.
+Then, in R⟨∂,
+�
+, E⟩, the operators L := ∂ + a and
+H := z ·
+�
+· z−1 satisfy
+L · H = 1.
+Proof. The same computation as in the commutative case above can be done
+also for noncommutative R without any changes. Note that zz−1 = 1 was the
+only property of z−1 used there, so it suffices that z−1 is a right inverse of z.
+Observe that from this statement over an abstract integro-differential ring
+(R, ∂,
+�
+), which was proven without referring to matrices at all, the analytic
+statement follows for arbitrary size n of the matrix A(x), provided we assume
+sufficient regularity of the functions involved.
+Based on Theorem 5.3, we now can complete the proof of Theorem 5.1 for
+arbitrary n ≥ 3. Before doing so, we first detail the required translation from
+28
+
+a first-order system to the scalar equation entirely at the operator level. For
+shorter notation, in the following, we again use the symbol ∂ also for the matrix
+diag(∂, . . . , ∂) ∈ R⟨∂,
+�
+, E⟩n×n.
+Lemma 5.4. Let (R, ∂,
+�
+) be an integro-differential ring and let
+L = ∂n +
+n−1
+�
+i=0
+ai · ∂i ∈ R⟨∂,
+�
+, E⟩
+with a0, . . . , an−1 ∈ R. Moreover, let H ∈ R⟨∂,
+�
+, E⟩n×n satisfy (∂+A)·H = In
+in R⟨∂,
+�
+, E⟩n×n, where
+A :=
+
+
+
+
+
+
+
+
+0
+−1
+0
+· · ·
+0
+...
+...
+...
+...
+...
+...
+...
+...
+0
+0
+· · ·
+· · ·
+0
+−1
+a0
+a1
+· · ·
+· · ·
+an−1
+
+
+
+
+
+
+
+
+.
+Then, the upper right entry H1,n of H satisfies L · H1,n = 1 in R⟨∂,
+�
+, E⟩ and
+Hi,n = ∂i−1 · H1,n for i ∈ {1, . . . , n}.
+Proof. For n = 1, the statement is trivial. So, we let n ≥ 2 in the following. In
+short, starting from the identity (∂ + A)·H = In of n× n matrices of operators,
+we multiply both sides from the left with a suitable S ∈ R⟨∂,
+�
+, E⟩n×n and
+inspect the last column. In particular, for elimination in the last n − 1 columns
+of ∂ + A, we use
+S :=
+
+
+
+
+
+
+
+
+
+
+
+1
+0
+· · ·
+· · ·
+· · ·
+0
+∂
+...
+...
+...
+∂2
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+∂n−2
+· · ·
+∂2
+∂
+1
+0
+s1
+· · ·
+· · ·
+· · ·
+sn−1
+1
+
+
+
+
+
+
+
+
+
+
+
+with sj := ∂n−j+�n−j−1
+i=0
+ai+j ·∂i ∈ R⟨∂,
+�
+, E⟩ for j ∈ {1, . . ., n−1}. Observing
+that, in R⟨∂, � , E⟩, we have −sj + sj+1 · ∂ = −aj for j ∈ {1, . . ., n − 2}, we
+29
+
+compute S · (∂ + A) explicitly:
+S ·
+
+
+
+
+
+
+
+
+∂
+−1
+0
+· · ·
+0
+0
+...
+...
+...
+...
+...
+...
+...
+...
+0
+0
+· · ·
+0
+∂
+−1
+a0
+a1
+· · ·
+an−2
+∂ + an−1
+
+
+
+
+
+
+
+
+=
+
+
+
+
+
+
+
+
+∂
+−1
+0
+· · ·
+0
+∂2
+0
+−1
+...
+...
+...
+...
+...
+...
+0
+∂n−1
+0
+· · ·
+0
+−1
+s1 · ∂ + a0
+0
+· · ·
+0
+−sn−1 + ∂ + an−1
+
+
+
+
+
+
+
+
+=
+
+
+
+
+
+
+
+
+
+∂
+−1
+0
+· · ·
+0
+∂2
+0
+−1
+...
+...
+...
+...
+...
+...
+0
+∂n−1
+...
+...
+−1
+L
+0
+· · ·
+· · ·
+0
+
+
+
+
+
+
+
+
+
+.
+Altogether, we obtain that
+
+
+
+
+
+
+
+
+
+∂
+−1
+0
+· · ·
+0
+∂2
+0
+−1
+...
+...
+...
+...
+...
+...
+0
+∂n−1
+...
+...
+−1
+L
+0
+· · ·
+· · ·
+0
+
+
+
+
+
+
+
+
+
+· H = S · (∂ + A) · H = S,
+where comparison of the entries in the last column of the left hand side and
+right hand side yields ∂ · H1,n − H2,n = 0, . . . , ∂n−1 · H1,n − Hn,n = 0 and
+L · H1,n = 0.
+Finally, we proceed with the proof of Theorem 5.1. Recalling the assump-
+tions, we fix a commutative integro-differential ring (R, ∂,
+�
+) and we let L =
+∂n+�n−1
+i=0 ai·∂i ∈ R⟨∂,
+�
+, E⟩ with a0, . . . , an−1 ∈ R. We also let z1, . . . , zn ∈ R
+such that Lzi = 0 and such that w := W(z1, . . . , zn) has a multiplicative inverse
+1
+w ∈ R.
+Proof of Theorem 5.1. Let n ≥ 3. In the (noncommutative) integro-differential
+30
+
+ring (Rn×n, ∂,
+�
+), we consider the matrices
+A :=
+
+
+
+
+
+
+
+
+0
+−1
+0
+· · ·
+0
+...
+...
+...
+...
+...
+...
+...
+...
+0
+0
+· · ·
+· · ·
+0
+−1
+a0
+a1
+· · ·
+· · ·
+an−1
+
+
+
+
+
+
+
+
+and
+Z :=
+
+
+
+
+
+z1
+· · ·
+zn
+∂z1
+· · ·
+∂zn
+...
+...
+∂n−1z1
+· · ·
+∂n−1zn
+
+
+
+
+
+and note that (∂ + A)Z = 0 and that Z−1 ∈ Rn×n exists, since det(Z) = w
+was assumed to be invertible in R.
+Then, we apply Theorem 5.3 to obtain
+(∂ + A) · (Z ·
+�
+· Z−1) = 1 in Rn×n⟨∂,
+�
+, E⟩.
+Instead of integro-differential
+operators with matrix coefficients in Rn×n, we now consider the objects as
+n × n matrices whose entries are integro-differential operators with coefficients
+in R, cf. Lemma 5.2. Then, by Lemma 5.4, we obtain that L·(Z ·� ·Z−1)1,n = 1
+holds in R⟨∂,
+�
+, E⟩. In order to compute the entry (Z ·
+�
+·Z−1)1,n, we can easily
+determine a general form of the entries of the matrix product Z ·
+�
+· Z−1 ∈
+R⟨∂,
+�
+, E⟩n×n:
+Z ·
+
+
+
+
+
+
+�
+0
+· · ·
+0
+0
+...
+...
+...
+...
+...
+...
+0
+0
+· · ·
+0
+�
+
+
+
+
+
+
+· Z−1 = Z ·
+
+
+
+�
+· (Z−1)1,1
+· · ·
+�
+· (Z−1)1,n
+...
+...
+�
+· (Z−1)n,1
+· · ·
+�
+· (Z−1)n,n
+
+
+
+=
+� n
+�
+k=1
+(∂i−1zk) · � · (Z−1)k,j
+�
+i,j=1,...,n
+.
+Then, we compute all entries
+(Z−1)i,n = (−1)n+i W(z1, . . . , zi−1, zi+1, . . . , zn)
+w
+in the last column of Z−1 ∈ Rn×n via Cramer’s rule (or via the cofactor matrix)
+so that we recognize H ∈ R⟨∂,
+�
+, E⟩ defined in the statement of Theorem 5.1
+as the top right entry (Z ·
+�
+· Z−1)1,n of the above matrix. This concludes the
+proof that L · H = 1.
+6
+Generalizing identities from calculus
+The generalizations of well-known identities presented in this section introduce
+additional terms involving the induced evaluation, which vanish if the evaluation
+is multiplicative. As explained in the previous section, the proofs mostly rely
+on computing irreducible forms for IDO.
+31
+
+6.1
+Initial value problems
+Recall that the formula given in (23) provides a particular solution of the inho-
+mogeneous linear ODE
+y(n)(x) + an−1(x)y(n−1)(x) + · · · + a0(x)y(x) = f(x).
+Moreover, this particular solution also satisfies the homogeneous initial condi-
+tions y(x0) = 0, y′(x0) = 0, . . . , y(n−1)(x0) = 0, see e.g. Theorem 6.4 in Chap-
+ter 3 of [6].
+The induced evaluation of an integro-differential ring allows us
+to model such properties algebraically. Similarly to the general proof of Theo-
+rem 5.1, we first investigate the general solution formula for homogeneous initial
+value problems of first-order systems. To this end, we fix a (not necessarily com-
+mutative) integro-differential ring R and we work in the corresponding ring of
+IDO R⟨∂,
+�
+, E⟩.
+As a general principle, not only in the ring R⟨∂,
+�
+, E⟩ but in any ring with
+unit element, if some H is a right inverse of some L and B, P are such that
+L · P = 0 and B · P = B, then G := (1 − P) · H satisfies
+L · G = 1
+and
+B · G = 0.
+So, by choosing an appropriate operator P ∈ R⟨∂,
+�
+, E⟩ that satisfies (∂+a)·P =
+0 and E · P = E, we can obtain the following version of Theorem 5.3 that also
+solves the homogeneous initial condition.
+Theorem 6.1. Let (R, ∂,
+�
+) be an integro-differential ring and let L = ∂ + a ∈
+R⟨∂,
+�
+, E⟩ with a ∈ R. Assume that z ∈ R is such that ∂z + az = 0 and in
+addition to a multiplicative right inverse z−1 ∈ R also a right inverse (Ez)−1 ∈ C
+exists. Then, in R⟨∂, � , E⟩, the operator
+G := (1 − z(Ez)−1 · E) · z ·
+�
+· z−1
+satisfies L · G = 1 and E · G = 0.
+Proof. By Theorem 5.3, we have that H := z ·
+�
+· z−1 satisfies L · H = 1. Now,
+letting P := z(Ez)−1 · E, we compute the normal forms of L · P and E · P:
+(∂ + a) · z(Ez)−1 · E =
+�
+z(Ez)−1 · ∂ + ∂z(Ez)−1�
+· E + az(Ez)−1 · E
+= z(Ez)−1 · ∂ · E = 0,
+E · z(Ez)−1 · E = Ez(Ez)−1 · E = E.
+Then, from L · P = 0 and E · P = E, it follows straightforwardly that G =
+(1 − P) · H has the properties claimed.
+Remark 6.2. If the evaluation E is multiplicative, then the existence of (Ez)−1 ∈
+C follows form the existence of z−1 ∈ R and, with (22), we also obtain E·z ·
+�
+=
+(Ez)E ·
+�
+= 0, which implies P · H = 0 and hence G = H. Therefore, the right
+inverse H obtained in Theorem 5.3 satisfies the initial value condition E ·H = 0
+already.
+32
+
+For conversion to the scalar case, we need to compute the element in the top
+right corner of G = (1 − Z(EZ)−1 · E) · Z ·
+���
+· Z−1 and show that it has the
+desired properties.
+Theorem 6.3. Let (R, ∂,
+�
+) be a commutative integro-differential ring and let
+L = ∂n + �n−1
+i=0 ai · ∂i ∈ R⟨∂,
+�
+, E⟩, n ≥ 1, with a0, . . . , an−1 ∈ R. Assume
+that z1, . . . , zn ∈ R are such that Lzi = 0 and w := W(z1, . . . , zn) has a mul-
+tiplicative inverse
+1
+w ∈ R. Moreover, assume that there are ci,j ∈ C such that
+E∂k �n
+i=1 ci,jzi = δj,k for all j, k ∈ {0, . . . , n − 1}. Then, with these ci,j and
+G :=
+n
+�
+k=1
+(−1)n−k
+�
+zk −
+n
+�
+i,j=1
+zici,j−1 · E · (∂j−1zk)
+�
+·
+�
+· W(z1,...,zk−1,zk+1,...,zn)
+w
+we have that L · G = 1 and E · G = 0 in R⟨∂,
+�
+, E⟩.
+Proof. As in the proof of Theorem 5.1, we consider the matrices
+A :=
+
+
+
+
+
+
+
+
+0
+−1
+0
+· · ·
+0
+...
+...
+...
+...
+...
+...
+...
+...
+0
+0
+· · ·
+· · ·
+0
+−1
+a0
+a1
+· · ·
+· · ·
+an−1
+
+
+
+
+
+
+
+
+and
+Z :=
+
+
+
+
+
+z1
+· · ·
+zn
+∂z1
+· · ·
+∂zn
+...
+...
+∂n−1z1
+· · ·
+∂n−1zn
+
+
+
+
+
+and note that (∂ + A)Z = 0 and that Z−1 ∈ Rn×n exists, since det(Z) = w was
+assumed to be invertible in R. Furthermore, we note that
+
+
+
+c1,0
+· · ·
+c1,n−1
+...
+...
+cn,0
+· · ·
+cn,n−1
+
+
+
+is the multiplicative (right) inverse of EZ in Cn×n. By Theorem 6.1, we conclude
+that ˜G := (1−Z(EZ)−1 ·E)·Z ·
+�
+·Z−1 ∈ Rn×n⟨∂,
+�
+, E⟩ satisfies (∂ +A)· ˜G = 1
+and E · ˜G = 0. Passing to R⟨∂,
+�
+, E⟩n×n via Lemma 5.2, the latter identity
+implies E · ˜Gi,j = 0 and the former implies L · ˜G1,n = 1 and ˜Gi,n = ∂i−1 · ˜G1,n
+by Lemma 5.4. Hence, we also have E · ∂i−1 · ˜G1,n = 0 for i ∈ {1, . . . , n}.
+Finally, we verify that ˜G1,n = G. With H := Z ·
+�
+· Z−1 ∈ R⟨∂,
+�
+, E⟩n×n,
+we have ˜G1,n = H1,n − �n
+i,j=1 Z1,i((EZ)−1)i,j · E · Hj,n. From the proof of
+Theorem 5.1, we obtain that Hj,n = (−1)n+k(∂j−1zk)·
+�
+· W(z1,...,zi−1,zi+1,...,zn)
+w
+.
+Altogether, this yields ˜G1,n = G.
+6.2
+Taylor formula
+Usually, Taylor’s theorem is only considered for sufficiently smooth functions.
+In integro-differential rings, an analog of the Taylor formula with integral re-
+mainder term
+f(x) =
+n
+�
+k=0
+f (k)(x0)
+k!
+xk +
+� x
+x0
+(x − t)n
+n!
+f (n+1)(t) dt
+33
+
+can be formulated. While the formula arising from the identity of operators in
+Theorem 6.6 is more complicated, it is also valid if singularities are present.
+We start by giving a first version of the Taylor formula where the remainder
+term is given as repeated integral. It simply follows by iterating (3) resp. (13).
+See also Corollary 2.2.1 in [23].
+Lemma 6.4. Let (R, ∂,
+�
+) be an integro-differential ring. In R⟨∂,
+�
+, E⟩ we have
+for any n ∈ N that
+1 =
+n
+�
+i=0
+� i · E · ∂i +
+� n+1 · ∂n+1.
+Proof. For any n ∈ N, we can use (13) to rewrite the right hand side:
+n
+�
+i=0
+� i · E · ∂i +
+� n+1 · ∂n+1 =
+n
+�
+i=0
+� i · E · ∂i +
+� n · (1 − E) · ∂n
+=
+n−1
+�
+i=0
+� i · E · ∂i +
+� n · ∂n.
+Setting n = 0 here gives E+
+�
+·∂ = 1. Hence, the claim follows by induction.
+Using (15) and the related identity in (20), we can always write operators
+� i · E as xi · E, where xi := � i1 as in Theorem 3.2. By (19) and (21), we can
+always write
+� n+1 without higher powers of
+�
+. To make the resulting expressions
+simpler, we restrict to the case that E is multiplicative on polynomials, i.e.
+Exmxn = 0 for all m, n ≥ 1.
+Lemma 6.5. Let (R, ∂,
+�
+) be an integro-differential ring such that the induced
+evaluation E satisfies Exmxn = 0 for all m, n ≥ 1. Then, in R⟨∂,
+�
+, E⟩, we
+have for any n ∈ N that
+� n+1 =
+n
+�
+k=0
+(−1)n−kxk ·
+�
+· xn−k −
+n−1
+�
+k=0
+n−k
+�
+j=1
+(−1)n−k−jxk · E · xj ·
+�
+· xn−k−j.
+Proof. We prove this identity by induction on n ∈ N. For n = 0, the right
+hand side directly yields
+�
+in agreement with the left hand side. Assuming the
+identity holds for some n ∈ N, we multiply both sides by
+�
+from the left and we
+rewrite the right hand side using (19) and (15) to obtain
+� n+2 =
+n
+�
+k=0
+(−1)n−k�� xk · � − � · � xk − E · � xk · � �
+· xn−k
+−
+n−1
+�
+k=0
+n−k
+�
+j=1
+(−1)n−k−j�
+xk · E · xj ·
+�
+· xn−k−j.
+34
+
+We have xk+1xn−k =
+�n+1
+k+1
+�
+xn+1 by Theorem 3.2, so we can expand the right
+hand side into
+n
+�
+k=0
+(−1)n−kxk+1 ·
+�
+· xn−k +
+n
+�
+k=0
+(−1)n−k+1
+�n + 1
+k + 1
+��
+· xn+1
+−
+n
+�
+k=0
+(−1)n−kE · xk+1 · � · xn−k −
+n−1
+�
+k=0
+n−k
+�
+j=1
+(−1)n−k−jxk+1 · E · xj · � · xn−k−j.
+Exploiting �n
+k=0(−1)n−k+1�n+1
+k+1
+�
+= (−1)n+1 in the second sum, we can regroup
+terms to obtain the right hand side of the claimed identity for n + 1.
+This
+completes the induction.
+Altogether, we obtain the following identity in R⟨∂,
+�
+, E⟩ generalizing the
+usual Taylor formula with integral remainder term. This identity holds over any
+integro-differential ring in which the induced evaluation is multiplicative on the
+integro-differential subring generated by 1.
+Theorem 6.6 (Taylor formula). Let (R, ∂,
+�
+) be an integro-differential ring
+such that E is multiplicative on the integro-differential subring generated by 1,
+i.e. Exmxn = 0 for all m, n ≥ 1. Then, for all f ∈ R and all n ∈ N we have
+1 =
+n
+�
+k=0
+xk · E · ∂k +
+n
+�
+k=0
+(−1)n−kxk ·
+�
+· xn−k · ∂n+1
+−
+n−1
+�
+k=0
+n−k
+�
+j=1
+(−1)n−k−jxk · E · xj · � · xn−k−j · ∂n+1
+Proof. Follows from Lemma 6.4 using
+� i·E = xi·E and Lemma 6.5, as explained
+above.
+In particular, if in addition to the assumptions of the theorem we have
+Q ⊆ R, then, with operators acting on some f, we have that
+f =
+n
+�
+k=0
+xk
+1
+k! E∂kf +
+n
+�
+k=0
+(−1)n−k
+k!(n − k)!xk
+1
+�
+xn−k
+1
+∂n+1f
+−
+n−1
+�
+k=0
+n−k
+�
+j=1
+(−1)n−k−j
+k!j!(n − k − j)!xk
+1Exj
+1
+�
+xn−k−j
+1
+∂n+1f
+While the first and the second sum correspond to the Taylor polynomial and the
+integral remainder term, the third sum corresponds to an additional polynomial
+that arises from our general setting allowing non-multiplicative evaluations. It
+can be viewed as an integro-differential algebraic version of the analytic formula
+−
+n−1
+�
+k=0
+(x − x0)k
+k!
+�� x
+x0
+(x − t)n−k − (x0 − t)n−k
+(n − k)!
+f (n+1)(t)dt
+�
+x=x0
+,
+35
+
+which vanishes for smooth functions f(x). Although in practice these additional
+terms yield zero even for many elements f that model singular functions in con-
+crete integro-differential rings, they cannot be dropped in general, as illustrated
+in C[x, x−1, ln(x)] by considering n = 1 and f = ln(x), for example. With in-
+tegration defined as in Example 2.13, we have x1 = x and with f = ln(x) the
+Taylor polynomial Ef + xE∂f vanishes. By ∂2f = − 1
+x2 , the second sum yields
+−
+�
+x∂2f + x
+�
+∂2f = ln(x) + 1 and the third sum −Ex
+�
+∂2f = −1 compensates
+the constant term. More generally, we can characterize the integro-differential
+rings where the additional polynomial does not play a role in the Taylor formula.
+Corollary 6.7. Let (R, ∂,
+�
+) be an integro-differential ring such that E is mul-
+tiplicative on the integro-differential subring generated by 1, i.e. Exmxn = 0 for
+all m, n ≥ 1. Then, we have Exng = 0 for all n ≥ 1 and g ∈ R if and only if
+we have
+f =
+n
+�
+k=0
+xkE∂kf +
+n
+�
+k=0
+(−1)n−kxk
+�
+xn−k∂n+1f
+for all n ∈ N and f ∈ R.
+Proof. If Exng = 0 for all n ≥ 1 and g ∈ R, then we have in particular
+Exj
+�
+xn−k−j∂n+1f = 0 for all j, k, n ∈ N and f ∈ R with 1 ≤ j ≤ n − k.
+Hence, Theorem 6.6 implies the claimed identity for all n ∈ N and f ∈ R.
+For the converse, let n ≥ 1 be minimal such that Exng ̸= 0 for some g ∈ R.
+With such g, we let f :=
+� ng and, by minimality of n, we obtain that
+n−1
+�
+k=0
+n−k
+�
+j=1
+(−1)n−k−jxkExj
+�
+xn−k−j∂n+1f = Exn
+�
+∂g = Exng − ExnEg = Exng
+is nonzero. Consequently, Theorem 6.6 implies that the claimed identity does
+not hold for this n and f.
+Acknowledgements
+This work was supported by the Austrian Science Fund (FWF): P 27229,
+P 31952, and P 32301. Part of this work was done while both authors were
+at the Radon Institute of Computational and Applied Mathematics (RICAM)
+of the Austrian Academy of Sciences. The authors would like to thank Alban
+Quadrat for bringing the book [23] to the attention of the second author during
+his stay at INRIA Saclay.
+A
+Normal forms for IDO in tensor rings
+The goal of this appendix is to state and prove a refinement of Theorem 4.2
+providing uniqueness of normal forms. Uniqueness is achieved by representing
+operators by elements of a tensor ring, which is formed on a module of basic
+operators generated by R, ∂,
+�
+, E. The ring of operators can be constructed
+36
+
+as quotient of the tensor ring, where relations of basic operators are encoded by
+tensor reduction rules. The main technical tool for proving uniqueness of nor-
+mal forms is a generalization of Bergman’s Diamond Lemma in tensor rings [3].
+For the convenience of the reader, we give a formal and largely self-contained
+summary of tensor reduction systems and we explain the translation of identi-
+ties of operators into this framework. In this appendix, K denotes a ring (not
+necessarily commutative) with unit element.
+In Section A.1, we start by recalling basic properties of bimodules and tensor
+rings on them. For further details on tensor rings and proofs see, for example,
+[34, 7]. Then, we recall decompositions with specialization, which are used for
+defining reduction rules. In Section A.2, we state the Diamond Lemma for tensor
+reduction systems with specialization from [14] and provide a summary of the
+relevant notions. In Section A.3, we construct an appropriate tensor ring along
+with a tensor reduction system for dealing with IDO. We use these to state The-
+orem A.3, which provides a precise formulation of uniqueness of normal forms
+of IDO in terms of tensors. In Section A.4, we give a complete proof of the
+theorem, where the necessary computations for verifying uniqueness of normal
+forms in the tensor ring are done in an automated way by our package TenReS in
+the computer algebra system Mathematica. These computations are contained
+in the Mathematica file accompanying this paper, which includes a log of the re-
+duction steps and is available at http://gregensburger.com/softw/tenres/.
+The theorem proved in that section even covers more general rings of operators,
+which allow to deal with additional functionals besides the induced evaluation.
+A.1
+Tensor rings on bimodules and decompositions
+A K-bimodule is a left K-module M which is also a right K-module satisfying
+the associativity condition (km)l = k(ml) for all m ∈ M and k, l ∈ K. A ring R
+that is a K-bimodule such that (xy)z = x(yz) for any x, y, z in R or K is called
+a K-ring. In particular, if K is a subring of some ring R, then R is a K-ring.
+We first recall basic properties of the tensor product on K-bimodules. For
+K-bimodules M, N, their K-tensor product M ⊗ N is a K-bimodule generated
+by the pure tensors {m ⊗ n | m ∈ M, n ∈ N} with relations
+(m + ˜m) ⊗ n = m ⊗ n + ˜m ⊗ n,
+m ⊗ (n + ˜n) = m ⊗ n + m ⊗ ˜n,
+and
+mk ⊗ n = m ⊗ kn
+having scalar multiplications
+k(m ⊗ n) = (km) ⊗ n
+and
+(m ⊗ n)k = m ⊗ (nk)
+for all m, ˜m ∈ M, n, ˜n ∈ N, and k ∈ K.
+We denote the tensor product of M with itself over K by M ⊗n = M ⊗· · ·⊗M
+(n factors). In particular, M ⊗1 = M and we interpret M ⊗0 as the K-module
+Kε, where ε denotes the empty tensor and right scalar multiplication satisfies
+(k1ε)k2 = (k1k2)ε for k1, k2 ∈ K. As a K-bimodule, the tensor ring K⟨M⟩ is
+37
+
+defined as the direct sum
+K⟨M⟩ =
+∞
+�
+n=0
+M ⊗n.
+It can be turned into a K-ring with unit element ε where multiplication M ⊗r ×
+M ⊗s → M ⊗(r+s) is defined via
+(m1 ⊗ · · · ⊗ mr, ˜m1 ⊗ · · · ⊗ ˜ms) �→ m1 ⊗ · · · ⊗ mr ⊗ ˜m1 ⊗ · · · ⊗ ˜ms.
+Via the tensor product, any decomposition of the module M carries over to
+a decomposition of the tensor ring K⟨M⟩. In particular, we use decompositions
+with specialization, which were introduced in [14]. These are given by a family
+(Mz)z∈Z of K-subbimodules of M and a subset X ⊆ Z with M = �
+z∈Z Mz =
+�
+x∈X Mx such that every module Mz, z ∈ Z, satisfies
+Mz =
+�
+x∈S(z)
+Mx
+where S(z) := {x ∈ X | Mx ⊆ Mz} is the set of specializations of z. Note that
+this definition implies S(x) = {x} for x ∈ X. For words W = w1 . . . wn in the
+word monoid ⟨Z⟩, we define the corresponding K-subbimodule of K⟨M⟩ by
+MW := Mw1 ⊗ · · · ⊗ Mwn.
+The notion of specialization extends from the alphabet Z to the whole word
+monoid ⟨Z⟩ by
+S(W) := {v1 . . . vn ∈ ⟨X⟩ | ∀i : vi ∈ S(wi)}
+such that S(W) = {V ∈ ⟨X⟩ | MV ⊆ MW }. We have the following generaliza-
+tion
+MW =
+�
+V ∈S(W)
+MV
+of the direct sum above and the decomposition
+K⟨M⟩ =
+�
+W∈⟨Z⟩
+MW =
+�
+W∈⟨X⟩
+MW
+of the tensor ring.
+A.2
+Tensor reduction systems with specialization
+Fixing a decomposition with specialization of M, a reduction rule for K⟨M⟩ is
+given by a pair r = (W, h) of a word W ∈ ⟨Z⟩ and a K-bimodule homomorphism
+h: MW → K⟨M⟩. It acts on tensors of the form a⊗w⊗b with a ∈ MA, w ∈ MW ,
+and b ∈ MB for some A, B ∈ ⟨Z⟩ by a ⊗ w ⊗ b →r a ⊗ h(w) ⊗ b. Later, we will
+specify homomorphisms h in concrete reduction rules (W, h) via their values on
+a generating set of MW . Formally, well-definedness of such homomorphisms can
+38
+
+be ensured by the universal property of the tensor product. A set Σ of such
+reduction rules is called a reduction system over Z on K⟨M⟩ and induces the
+two-sided reduction ideal
+IΣ := (t − h(t) | (W, h) ∈ Σ and t ∈ MW ) ⊆ K⟨M⟩.
+For computing in the factor ring K⟨M⟩/IΣ, we apply the reduction relation →Σ
+induced by Σ on K⟨M⟩. It reduces a tensor t ∈ K⟨M⟩ to a tensor s ∈ K⟨M⟩
+if there is an r ∈ Σ such that t →r s. We say that t can be reduced to s by Σ
+if t = s or there exists a finite sequence of reduction rules r1, . . . , rn in Σ such
+that
+t →r1 t1 →r2 · · · →rn−1 tn−1 →rn s.
+If one tensor can be reduced to another, then their difference is contained in
+IΣ and they represent the same element of K⟨M⟩/IΣ. The K-subbimodule of
+irreducible tensors
+K⟨M⟩irr =
+�
+W∈⟨X⟩irr
+MW
+can be characterized by the set of irreducible words ⟨X⟩irr ⊆ ⟨X⟩, which consists
+of those words that avoid subwords arising as specializations S(W) of words
+occurring in reduction rules (W, h) ∈ Σ. The irreducible tensors to which a
+given tensor t can be reduced, are called its normal forms. If t has a unique
+normal form, it is denoted by t↓Σ.
+An ambiguity is a minimal situation where two (not necessarily distinct) re-
+duction rules can be applied to tensors in different ways. For each pure tensor
+of this kind, the corresponding S-polynomial is the difference of the results of
+the two reduction steps. For example, an overlap ambiguity arises from two
+reduction rules (AB1, h), (B2C, ˜h) ∈ Σ, where A, B1, B2, C ∈ ⟨Z⟩ are nonempty
+such that B1, B2 are equal or have a common specialization, and corresponding
+S-polynomials are referred to by SP(AB1, B2C). An ambiguity is called resolv-
+able, if all its S-polynomials can be reduced to zero by Σ. If all ambiguities of Σ
+are resolvable, then the reduction relation induced by Σ on K⟨M⟩ is confluent
+and, by abuse of language, we also call Σ confluent. This means that there are
+no hidden consequences implied in K⟨M⟩/IΣ by the identities explicitly specified
+by Σ.
+The following theorem relies on the existence of a partial order of words in
+⟨Z⟩ that has certain properties, which are briefly explained now. A partial order
+≤ on ⟨Z⟩ is called a semigroup partial order if it is compatible with concatenation
+of words. If in addition the empty word ǫ is the least element of ⟨Z⟩, then ≤
+is called a monoid partial order. It is called Noetherian if there are no infinite
+descending chains. We call a partial order ≤ on ⟨Z⟩ consistent with specialization
+if every strict inequality V < W implies ˜V < ˜W for all specializations ˜V ∈ S(V )
+and ˜W ∈ S(W). A partial order ≤ on ⟨Z⟩ is compatible with a reduction system
+Σ over Z on K⟨M⟩ if for all (W, h) ∈ Σ the image of h is contained in the sum
+of modules MV where V ∈ ⟨Z⟩ satisfies V < W.
+39
+
+Theorem A.1. [14, Thm. 20] Let M be a K-bimodule and let (Mz)z∈Z be
+a decomposition with specialization.
+Let Σ be a reduction system over Z on
+K⟨M⟩ and let ≤ be a Noetherian semigroup partial order on ⟨Z⟩ consistent with
+specialization and compatible with Σ. Then, the following are equivalent:
+1. All ambiguities of Σ are resolvable.
+2. Every t ∈ K⟨M⟩ has a unique normal form t↓Σ.
+3. K⟨M⟩/IΣ and K⟨M⟩irr are isomorphic as K-bimodules.
+Moreover, if these conditions are satisfied, then we can define a multiplication
+on K⟨M⟩irr by s · t := (s ⊗ t)↓Σ so that K⟨M⟩/IΣ and K⟨M⟩irr are isomorphic
+as K-rings.
+A.3
+Tensor reduction systems for IDO
+Before using the tensor setting to construct the ring of integro-differential op-
+erators R⟨∂,
+�
+, E⟩, we illustrate this construction on the well-known ring of
+differential operators R⟨∂⟩ to highlight some of the special properties of the
+construction. Usually, the ring of differential operators with coefficients from R
+is constructed via skew polynomials �
+i fi∂i over R in one indeterminate ∂, with
+commutation rule ∂ · f = f · ∂ + ∂f. Let (R, ∂,
+�
+) be an integro-differential ring
+and let C denote its ring of constants. In the following, we consider K-tensor
+rings with K := C.
+Example A.2. The module of basic operators that generates all differential
+operators is given by
+M := MR ⊕ MD,
+with K-bimodules
+MR := R
+(24)
+and MD defined as the free left K-module
+MD := K∂
+(25)
+generated by the symbol ∂, which we view as a K-bimodule with the right
+multiplication c∂ · d = cd∂ for all c, d ∈ K. This definition is based on left
+K-linearity of the derivation ∂ on R. The commutation rule ∂ · f = f · ∂ + ∂f
+coming from the Leibniz rule in R translates to the tensor reduction rule
+(DR, ∂⊗f �→ f⊗∂ + ∂f).
+This rule is formalized by the K-bimodule homomorphism MDR = MD ⊗ MR →
+K⟨M⟩ defined by ∂⊗f �→ f⊗∂ + ∂f on tensors ∂⊗f generating MDR. Note that
+this homomorphism represents a parameterized family of identities. Reduction
+of tensors by the rule above allows to syntactically move all multiplication op-
+erators to the left of any differentiation operator.
+40
+
+Note that, in order to correctly model differential operators as equivalence
+classes of tensors in K⟨M⟩, other relations among operators need to be phrased
+as tensor reduction rules as well. This is because the tensor ring K⟨M⟩ itself
+is constructed without respecting relations coming from multiplication in R.
+For instance, the composition of two multiplication operators f and g is a mul-
+tiplication operator again, which leads to the reduction rule (RR, f⊗g �→ fg)
+defined on the module MRR = MR ⊗ MR. Moreover, the multiplication operator
+that multiplies by 1 acts like the identity operator, which is represented by the
+empty tensor ε. To define reduction rules that act only on C instead of all of R,
+we need a direct decomposition of the K-bimodule
+MR = MK ⊕ M˜R,
+(26)
+which in our case can be given by
+MK := K
+and
+M˜R :=
+�
+R
+(27)
+based on Lemma 2.3. Then, we can define a reduction rule on MK = C by 1 �→ ε.
+Altogether, the tensor reduction system ΣDiff = {rK, rRR, rDR} for differential
+operators is given by the three reduction rules
+{(K, 1 �→ ε),
+(RR, f⊗g �→ fg),
+(DR, ∂⊗f �→ f⊗∂ + ∂f)}
+defined above. It induces the two-sided ideal
+IDiff := (t − h(t) | (W, h) ∈ ΣDiff and t ∈ MW )
+= (1 − ε, f⊗g − fg, ∂⊗f − f⊗∂ − ∂f | f, g ∈ R)
+in the ring K⟨M⟩ and computations with differential operators are modelled in
+the quotient ring
+R⟨∂⟩ := K⟨M⟩/IDiff.
+Tensors that are not reducible w.r.t. ΣDiff are precisely K-linear combinations
+of pure tensors of the form ∂⊗i and f ⊗ ∂⊗i, where i ∈ N0 and f ∈
+�
+R. With
+alphabets X = {K, ˜R, D} and Z = X ∪{R}, one can check that all ambiguities of
+ΣDiff are resolvable. Using an appropriate ordering of words, one can show that
+the conditions of Theorem A.1 are satisfied by this construction of R⟨∂⟩.
+Proceeding to an analogous construction of the ring R⟨∂,
+�
+, E⟩ from Defini-
+tion 4.1, we also require tensor reduction rules corresponding to the identities
+(12)–(16), in addition to the three rules from the example above. To this end,
+analogous to MD above, we introduce the K-bimodules
+MI := K
+�
+and
+ME := KE,
+(28)
+which are freely generated as left K-modules by the symbols
+�
+and E, respec-
+tively. Then, we consider the K-tensor ring on the K-bimodule
+M := MR ⊕ MD ⊕ MI ⊕ ME.
+41
+
+K
+1 �→ ε
+ID
+�
+⊗∂ �→ ε − E
+RR
+f⊗g �→ fg
+DRE
+∂⊗f⊗E �→ ∂f⊗E
+DR
+∂⊗f �→ f⊗∂ + ∂f
+IRE
+�
+⊗f⊗E �→
+�
+f⊗E
+DI
+∂⊗
+�
+�→ ε
+ERE
+E⊗f⊗E �→ (Ef)E
+Table 2: Defining reduction system for integro-differential operators
+Altogether, we obtain the tensor reduction system given in Table 2. The two-
+sided ideal IIDO induced by it allows to construct the ring R⟨∂,
+�
+, E⟩ as the
+quotient K⟨M⟩/IIDO. However, this reduction system is not confluent. In order
+to obtain normal forms that are unique as tensors in K⟨M⟩, we need a confluent
+tensor reduction system on K⟨M⟩ that induces the same ideal IIDO. The con-
+fluent tensor reduction system given in Table 3 can be obtained by turning the
+identities of Table 1 into tensor reduction rules and including the rules rK and
+rRR from above. This allows us to state the following more precise version of
+K
+1 �→ ε
+EI
+E⊗
+�
+�→ 0
+RR
+f⊗g �→ fg
+IRD
+� ⊗f⊗∂ �→ f − E⊗f − � ⊗∂f
+DR
+∂⊗f �→ f⊗∂ + ∂f
+IRE
+�
+⊗f⊗E �→
+�
+f⊗E
+DE
+∂⊗E �→ 0
+IRI
+�
+⊗f⊗
+�
+�→
+�
+f⊗
+�
+− E⊗
+�
+f⊗
+�
+−
+�
+⊗
+�
+f
+DI
+∂⊗
+�
+�→ ε
+ID
+�
+⊗∂ �→ ε − E
+ERE
+E⊗f⊗E �→ (Ef)E
+IE
+�
+⊗E �→
+�
+1⊗E
+EE
+E⊗E �→ E
+II
+�
+⊗
+�
+�→
+�
+1⊗
+�
+− E⊗
+�
+1⊗
+�
+−
+�
+⊗
+�
+1
+Table 3: Confluent reduction system ΣIDO for integro-differential operators
+Theorem 4.2. Note that multiples like f · ∂ are treated differently now, due to
+the fact that we are working in the tensor ring K⟨M⟩ and we have the reduction
+rule (K, 1 �→ ε), which splits f ∈ R according to Lemma 2.3.
+Theorem A.3. Let (R, ∂,
+�
+) be an integro-differential ring with constants K =
+C. Let M be given as above in terms of the modules defined in Eqs. (26), (27),
+(25), and (28) and let the tensor reduction system ΣIDO be defined by Table 3.
+Then every t ∈ K⟨M⟩ has a unique normal form t↓ΣIDO∈ K⟨M⟩, which can
+be written as a K-linear combination of pure tensors of the form
+f ⊗ ∂⊗j,
+f ⊗
+�
+⊗ g,
+f ⊗ E ⊗ g ⊗ ∂⊗j,
+or
+f ⊗ E ⊗ h ⊗
+�
+⊗ g
+where j ∈ N0, f, g, h ∈
+�
+R, and each f and g may be absent. Moreover,
+R⟨∂,
+�
+, E⟩ ∼= K⟨M⟩irr
+as K-rings, where multiplication on K⟨M⟩irr is defined by s · t := (s ⊗ t)↓ΣIDO.
+Instead of proving this theorem, we will prove a more general one below,
+which allows to include additional functionals into the construction of the ring
+of operators. Theorem A.3 follows from Theorem A.5 by specializing Φ = {E}.
+42
+
+A.4
+Proof of normal forms for IDO with functionals
+To treat more general problems than the initial value problems in Section 6.1, it
+is useful to include additional functionals into the ring of operators. For instance,
+dealing with boundary problems requires evaluations at more than one point.
+In general, we consider a set Φ of K-linear functionals R → K including E. We
+consider the K-bimodule MΦ defined as free left K-module
+MΦ := KΦ
+(29)
+generated by the elements of Φ, where we define right multiplication in terms
+of Φ by
+��
+ϕ∈Φ cϕϕ
+�
+· d = �
+ϕ∈Φ cϕdϕ for cϕ, d ∈ K.
+Note that K-linear
+combinations of K-linear maps are not necessarily K-linear again, if K is not
+commutative.
+Since ∂,
+�
+, and all elements of Φ are K-linear, we have, for
+instance, that ϕfψg = (ϕf)ψg for all f, g ∈ R and ϕ, ψ ∈ Φ.
+This allows
+to extend the last three reduction rules of Table 2 to cover all elements of Φ
+instead of the evaluation E only. Altogether, we consider the K-tensor ring on
+the K-bimodule
+M := MR ⊕ MD ⊕ MI ⊕ MΦ
+(30)
+and we have the defining reduction system given in Table 4.
+K
+1 �→ ε
+ID
+� ⊗∂ �→ ε − E
+RR
+f⊗g �→ fg
+DRΦ
+∂⊗f⊗ϕ �→ ∂f⊗ϕ
+DR
+∂⊗f �→ f⊗∂ + ∂f
+IRΦ
+�
+⊗f⊗ϕ �→
+�
+f⊗ϕ
+DI
+∂⊗
+�
+�→ ε
+ΦRΦ
+ϕ⊗f⊗ψ �→ (ϕf)ψ
+Table 4: Defining reduction system for integro-differential operators with func-
+tionals
+Definition A.4. Let (R, ∂,
+�
+) be an integro-differential ring with constants K =
+C and let Φ be a set of K-linear functionals R → K including E. Let the K-
+bimodule M be defined as above in Eqs. (24), (25), (28), (29), and (30). We
+call
+R⟨∂,
+�
+, Φ⟩ := K⟨M⟩/IIDOΦ
+the ring of integro-differential operators with functionals Φ, where IIDOΦ is the
+two-sided ideal induced by the reduction system obtained from Table 4 using also
+submodules of M defined in Eq. (27).
+We use Theorem A.1 above to determine unique normal forms of tensors.
+Again, the reduction system given by Table 4 is not confluent and we need
+to construct a confluent reduction system, like ΣIDOΦ given in Table 5, by a
+completion process similar to how Table 3 was obtained. Observe that, whenever
+Φ = {E}, the ring R⟨∂, � , Φ⟩ and the relation →ΣIDOΦ specialize to R⟨∂, � , E⟩
+and →ΣIDO, respectively.
+43
+
+K
+1 �→ ε
+EI
+E⊗
+�
+�→ 0
+RR
+f⊗g �→ fg
+IRD
+�
+⊗f⊗∂ �→ f − E⊗f −
+�
+⊗∂f
+DR
+∂⊗f �→ f⊗∂ + ∂f
+IRΦ
+�
+⊗f⊗ϕ �→
+�
+f⊗ϕ
+DΦ
+∂⊗ϕ �→ 0
+IRI
+�
+⊗f⊗
+�
+�→
+�
+f⊗
+�
+− E⊗
+�
+f⊗
+�
+−
+�
+⊗
+�
+f
+DI
+∂⊗
+�
+�→ ε
+ID
+�
+⊗∂ �→ ε − E
+ΦRΦ
+ϕ⊗f⊗ψ �→ (ϕf)ψ
+IΦ
+�
+⊗ϕ �→
+�
+1⊗ϕ
+ΦΦ
+ϕ⊗ψ �→ (ϕ1)ψ
+II
+�
+⊗
+�
+�→
+�
+1⊗
+�
+− E⊗
+�
+1⊗
+�
+−
+�
+⊗
+�
+1
+Table 5: Confluent reduction system ΣIDOΦ for integro-differential operators
+with functionals
+Theorem A.5. Let (R, ∂,
+�
+) be an integro-differential ring with constants K =
+C and let Φ be a set of K-linear functionals R → K including E. Let M and
+R⟨∂,
+�
+, Φ⟩ be defined as in Definition A.4 above and let the tensor reduction
+system ΣIDOΦ be defined by Table 5.
+Then every t ∈ K⟨M⟩ has a unique normal form t↓ΣIDOΦ∈ K⟨M⟩, which can
+be written as a K-linear combination of pure tensors of the form
+f ⊗ ∂⊗j,
+f ⊗
+�
+⊗ g,
+f ⊗ ϕ ⊗ h ⊗ ∂⊗j,
+or
+f ⊗ ϕ ⊗ h ⊗
+�
+⊗ g
+where j ∈ N0, f, g, h ∈
+�
+R, ϕ ∈ Φ, and each f, g, h may be absent such that
+ϕ ⊗ h ⊗
+�
+does not specialize to E ⊗
+�
+. Moreover,
+R⟨∂,
+�
+, Φ⟩ ∼= K⟨M⟩irr
+as K-rings, where multiplication on K⟨M⟩irr is defined by s · t := (s ⊗ t)↓ΣIDOΦ.
+Proof. We use the alphabets X := {K, ˜R, D, I, E, ˜Φ} and Z := X ∪ {R, Φ}, which
+turns (Mz)z∈Z into a decomposition with specialization for the module M,
+where S(R) = {K, ˜R} and S(Φ) = {E, ˜Φ}. For defining a Noetherian monoid
+partial order ≤ on ⟨Z⟩ consistent with specialization that is compatible with
+ΣIDOΦ, it is sufficient to require the order to satisfy
+DR > RD
+and
+I > E˜R.
+For instance, we could first define a monoid total order on ⟨{R, D, I, Φ}⟩ ⊆
+⟨Z⟩ by counting occurrences of the letter I and breaking ties with any degree-
+lexicographic order satisfying D > R and then generate from it a partial order
+on ⟨Z⟩ that is consistent with specialization.
+By our Mathematica package TenReS, we generate all ambiguities of ΣIDOΦ
+and verify that they are resolvable.
+There are 54 ambiguities and indeed
+all S-polynomials reduce to zero, see the accompanying Mathematica file at
+http://gregensburger.com/softw/tenres/. Here, we just give two short ex-
+amples. The first one illustrates in its last step of computation that also iden-
+44
+
+tities in MR, like the Leibniz rule, need to be used.
+SP(IRD, DR) = (f − E ⊗ f −
+�
+⊗ ∂f) ⊗ g −
+�
+⊗ f ⊗ (g ⊗ ∂ + ∂g)
+→rRR fg − E ⊗ fg −
+�
+⊗ (∂f)g −
+�
+⊗ fg ⊗ ∂ −
+�
+⊗ f∂g
+→rIRD −
+�
+⊗ (∂f)g +
+�
+⊗ ∂fg −
+�
+⊗ f∂g
+=
+�
+⊗ (−(∂f)g + ∂fg − f∂g) = 0
+The second one illustrates that ambiguities involving specialization also need to
+be considered.
+SP(ΦΦ, EI) = (ϕ1)E ⊗
+�
+− ϕ ⊗ 0 →rEI 0
+Since all ambiguities are resolvable, by Theorem A.1 every element in K⟨M⟩ has
+a unique normal form and R⟨∂,
+�
+, Φ⟩ ∼= K⟨M⟩irr as K-rings.
+It remains to determine the explicit form of elements in K⟨M⟩irr. In order to
+do so, we determine the set of irreducible words ⟨X⟩irr in ⟨X⟩. Irreducible words
+containing only the letters K, ˜R, E, ˜Φ have to avoid the subwords arising from
+the reduction rules K, S(RR) = {KK, K˜R, ˜RK, ˜R˜R}, S(ΦΦ) = {EE, E˜Φ, ˜ΦE, ˜Φ˜Φ},
+and S(ΦRΦ). Hence they are given by
+ǫ, ˜R, E, ˜Φ, ˜RE, ˜R˜Φ, E˜R, ˜Φ˜R, ˜RE˜R, ˜R˜Φ˜R
+Allowing also the letter D, we have to avoid the subwords coming from S(DR) =
+{DK, D˜R} and S(DΦ) = {DE, D˜Φ}. Therefore, we can only append words Dj
+with j ∈ N0 to the irreducible words determined so far, in order to obtain all
+elements of ⟨X⟩irr not containing the letter I. Finally, we also consider the letter
+I. Since we have to avoid the subwords S(IΦ) = {IE, I˜Φ}, ID, and II, any letter
+immediately following I has to be ˜R. In addition, we have to avoid the subwords
+S(IRΦ) = {IKE, IK˜Φ, I˜RE, I˜R˜Φ}, S(IRD) = {IKD, I˜RD}, and S(IRI) = {IKI, I˜RI},
+so the letter I cannot be followed by a subword of length greater than one.
+Therefore, the letter I can appear at most once in an element of ⟨X⟩irr and, since
+subwords EI and DI have to be avoided, it can only be immediately preceded by
+the letters ˜R or ˜Φ. Altogether, the elements of ⟨X⟩irr are precisely of the form
+˜RUDj
+or
+˜RV I˜R,
+where j ∈ N0 and each of ˜R and U ∈ {E, ˜Φ, E˜R, ˜Φ˜R} and V ∈ {˜Φ, E˜R, ˜Φ˜R} may
+be absent.
+Remark A.6. As discussed in Remark 2.12, the induced evaluation of an
+integro-differential ring often is multiplicative in concrete examples. Also when
+considering several point evaluations of regular functions, the resulting func-
+tionals are multiplicative, i.e. ϕfg = (ϕf)ϕg for all f, g ∈ R. We discuss how
+reflecting this additional property of some functionals in the ring of operators
+influences the normal forms of operators. To this end, we consider a subset
+Φm ⊆ {ϕ ∈ Φ | ϕ is multiplicative and ϕ1 = 1}
+45
+
+of Φ, which may or may not include the evaluation E. For the corresponding
+elements ϕ ∈ Φm in R⟨∂,
+�
+, Φ⟩, we impose
+ϕ · f = (ϕf)ϕ
+for all f ∈ R.
+To model this identity by reduction rules, we consider the
+submodule
+MΦm := KΦm
+of MΦ. Evidently, we have the decomposition
+MΦ = ME ⊕ M˜Φm ⊕ M˜Φ
+where the submodules M˜Φm and M˜Φ are generated by Φm \ {E} and Φ \ ({E} ∪
+Φm), respectively. We include the reduction rule
+(ΦmR, ϕ⊗f �→ (ϕf)ϕ)
+into Tables 4 and 5, where ϕ in the formula defining this K-bimodule homomor-
+phism on MΦmR is not a general element of MΦm but of Φm and the definition
+needs to be extended by left K-linearity to all of MΦmR.
+Theorem A.5 generalizes to this situation with the additional restriction on
+normal forms in K⟨M⟩ that h needs to be absent whenever ϕ ∈ Φm.
+The
+proof is analogous by adapting the alphabets and the order accordingly. The
+additional reduction rule gives rise to additional ambiguities, whose resolvability
+is also checked in the Mathematica file. Determination of irreducible words also
+needs to be adapted accordingly. The resulting theorem includes Theorem A.5
+for Φm = ∅ and it includes Theorem 27 from [14] for Φm = Φ, i.e. when all
+functionals are multiplicative.
+References
+[1] Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven,
+Asymptotic differential algebra and model theory of transseries, Annals of
+Mathematics Studies, vol. 195, Princeton University Press, Princeton, NJ,
+2017.
+[2] Vladimir V. Bavula, The algebra of integro-differential operators on a poly-
+nomial algebra, J. Lond. Math. Soc. (2) 83 (2011), 517–543.
+[3] George M. Bergman, The diamond lemma for ring theory, Adv. in Math.
+29 (1978), 178–218.
+[4] Bruno Buchberger, An algorithm for finding the bases elements of the
+residue class ring modulo a zero dimensional polynomial ideal (German),
+Ph.D. thesis, University of Innsbruck, 1965.
+46
+
+[5] Thomas Cluzeau, Jamal Hossein Poor, Alban Quadrat, Clemens G. Raab,
+and Georg Regensburger, Symbolic computation for integro-differential-
+time-delay operators with matrix coefficients, IFAC-PapersOnLine 51
+(2018), no. 14, 153–158.
+[6] Earl A. Coddington and Norman Levinson, Theory of ordinary differential
+equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955.
+[7] Paul M. Cohn, Further algebra and applications, Springer-Verlag London,
+Ltd., London, 2003.
+[8] Gerald A. Edgar, Transseries for beginners, Real Anal. Exchange 35 (2010),
+no. 2, 253–309.
+[9] Xing Gao, Li Guo, and Markus Rosenkranz, On rings of differential Rota-
+Baxter operators, Internat. J. Algebra Comput. 28 (2018), no. 1, 1–36.
+[10] Li Guo, An introduction to Rota-Baxter algebra, Surveys of Modern Math-
+ematics, vol. 4, International Press, Somerville, MA; Higher Education
+Press, Beijing, 2012.
+[11] Li Guo and William Keigher, On differential Rota-Baxter algebras, J. Pure
+Appl. Algebra 212 (2008), 522–540.
+[12] Li Guo, Georg Regensburger, and Markus Rosenkranz, On integro-
+differential algebras, J. Pure Appl. Algebra 218 (2014), 456–473.
+[13] Jamal Hossein Poor, Tensor reduction systems for rings of linear operators,
+Ph.D. thesis, Johannes Kepler University Linz, Austria, 2018.
+[14] Jamal Hossein Poor, Clemens G. Raab, and Georg Regensburger, Algo-
+rithmic operator algebras via normal forms in tensor rings, J. Symbolic
+Comput. 85 (2018), 247–274.
+[15] Irving Kaplansky, An introduction to differential algebra, Publ. Inst. Math.
+Univ. Nancago, No. 5, Hermann, Paris, 1957.
+[16] William F. Keigher, Adjunctions and comonads in differential algebra, Pa-
+cific J. Math. 59 (1975), no. 1, 99–112.
+[17]
+, On the ring of Hurwitz series, Comm. Algebra 25 (1997), no. 6,
+1845–1859.
+[18] William F. Keigher and F. Leon Pritchard, Hurwitz series as formal func-
+tions, J. Pure Appl. Algebra 146 (2000), no. 3, 291–304.
+[19] Donald E. Knuth and Peter B. Bendix, Simple word problems in universal
+algebras, Computational Problems in Abstract Algebra, Pergamon, Oxford,
+1970, pp. 263–297.
+47
+
+[20] Ernst E. Kummer, Ueber die Transcendenten, welche aus wiederholten Inte-
+grationen rationaler Formeln entstehen, J. Reine Angew. Math. 21 (1840),
+74–90, 193–225, and 328–371.
+[21] Ivan A. Lappo-Danilevski, Résolution algorithmique des problèmes réguliers
+de Poincaré et de riemann (Mémoire premier: Le problème de Poincaré,
+concernant de la construction d’un groupe de monodromie d’un système
+donné d’équations différentielles linéaires aux intégrales régulières), J. Soc.
+Phys.-Math. Léningrade 2 (1928), no. 1, 94–120.
+[22] Teo Mora, An introduction to commutative and noncommutative Gröbner
+bases, Theoret. Comput. Sci. 134 (1994), no. 1, 131–173.
+[23] Danuta Przeworska-Rolewicz, Algebraic analysis, PWN—Polish Scientific
+Publishers, Warsaw; D. Reidel Publishing Co., Dordrecht, 1988.
+[24]
+, Two centuries of the term “algebraic analysis”, Algebraic analysis
+and related topics (Warsaw, 1999), Banach Center Publ., vol. 53, Polish
+Acad. Sci. Inst. Math., Warsaw, 2000, pp. 47–70.
+[25] Alban Quadrat, A constructive algebraic analysis approach to Artstein’s
+reduction of linear time-delay systems, IFAC-PapersOnLine 48 (2015),
+no. 12, 209–214.
+[26] Alban Quadrat and Georg Regensburger, Computing polynomial solutions
+and annihilators of integro-differential operators with polynomial coeffi-
+cients, Algebraic and symbolic computation methods in dynamical systems,
+Adv. Delays Dyn., vol. 9, Springer, Cham, 2020, pp. 87–114.
+[27] Rimhak Ree, Lie elements and an algebra associated with shuffles, Ann. of
+Math. (2) 68 (1958), 210–220.
+[28] Georg Regensburger, Markus Rosenkranz, and Johannes Middeke, A skew
+polynomial approach to integro-differential operators, Proceedings of ISSAC
+’09 (New York, NY, USA), ACM, 2009, pp. 287–294.
+[29] Markus Rosenkranz, A new symbolic method for solving linear two-point
+boundary value problems on the level of operators, J. Symbolic Comput. 39
+(2005), 171–199.
+[30] Markus Rosenkranz and Georg Regensburger, Solving and factoring bound-
+ary problems for linear ordinary differential equations in differential alge-
+bras, J. Symbolic Comput. 43 (2008), 515–544.
+[31] Markus Rosenkranz, Georg Regensburger, Loredana Tec, and Bruno
+Buchberger, Symbolic analysis for boundary problems: From rewriting to
+parametrized Gröbner bases, Numerical and Symbolic Scientific Computing:
+Progress and Prospects (Ulrich Langer and Peter Paule, eds.), Springer Vi-
+enna, Vienna, 2012, pp. 273–331.
+48
+
+[32] Markus Rosenkranz and Nitin Serwa, An integro-differential structure for
+Dirac distributions, J. Symbolic Comput. 92 (2019), 156–189.
+[33] Gian-Carlo Rota, Twelve problems in probability no one likes to bring up,
+Algebraic combinatorics and computer science, Springer Italia, Milan, 2001,
+pp. 57–93.
+[34] Louis H. Rowen, Ring theory, student ed., Academic Press, Inc., Boston,
+MA, 1991.
+[35] Richard P. Stanley, Differentiably finite power series, European J. Combin.
+1 (1980), no. 2, 175–188.
+[36] Joris van der Hoeven, Transseries and real differential algebra, Lecture
+Notes in Mathematics, vol. 1888, Springer-Verlag, Berlin, 2006.
+49
+