Appell and Sheffer Sequences

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Comptes Rendus Mathématique

Sergio A. Carrillo and Miguel Hurtado Appell and Sheﬀer sequences: on their characterizations through functionals and examples Volume 359, issue 2 (2021), p. 205-217.

This article is licensed under the Creative Commons Attribution 4.0 International License. http://creativecommons.org/licenses/by/4.0/

Les Comptes Rendus. Mathématique sont membres du Centre Mersenne pour l’édition scientiﬁque ouverte www.centre-mersenne.org Comptes Rendus Mathématique 2021, 359, n 2, p. 205-217 https://doi.org/10.5802/crmath.172

Theory of functions / Théorie des fonctions

Appell and Sheﬀer sequences: on their characterizations through functionals and examples

Sergio A. Carrilloa and Miguel Hurtadoa

a Programa de matemáticas, Universidad Sergio Arboleda, Calle 74 # 14-14, Bogotá, Colombia. E-mails: [email protected], [email protected]

Abstract. The aim of this paper is to present a new simple recurrence for Appell and Sheffer sequences in terms of the linear functional that defines them, and to explain how this is equivalent to several well-known characterizations appearing in the literature. We also give several examples, including integral representations of the inverse operators associated to Bernoulli and Euler polynomials, and a new integral representation of the re-scaled Hermite d-orthogonal polynomials generalizing the Weierstrass operator related to the Hermite polynomials. Résumé. L’objectif de cet article est de présenter une nouvelle récurrence simple pour les suites d’Appell et de Sheffer en termes de la fonctionnelle linéaire qui les définit, et d’expliquer comment cela équivaut à plusieurs caractérisations bien connues qui apparaissent dans la littérature. Nous donnons aussi plusieurs exemples, y compris des représentations intégrales des opérateurs inverses associés aux polynômes de Bernoulli et d’Euler, et une nouvelle représentation intégrale des polynômes d’Hermite d-orthogonaux remis à l’échelle, qui généralise l’opérateur de Weierstrass associé aux polynômes d’Hermite. Keywords. Sheffer and Appell sequences, Bernoulli, Euler and Hermite d-orthogonal polynomials. 2020 Mathematics Subject Classification. 05A40, 11B83, 11B68. Funding. First author is supported by Ministerio de Economía y Competitividad from Spain, under the Project “Métodos asintóticos, algebraicos y geométricos en foliaciones singulares y sistemas dinámicos” (Ref.: PID2019-105621GB-I00) and Univ. Sergio Arboleda project IN.BG.086.20.002.

Manuscript received 19th October 2020, revised 11th December 2020, accepted 17th December 2020.

1. Introduction

A remarkable class of polynomials are the Appell sequences having applications in Number theory, Probability, and the theory of functions. They vastly generalize monomials arising naturally from Taylor’s formula with integral rest, see [10, Chapitre VI] [11, Appendix A] and they include famous polynomial sequences. For instance, the Bernoulli polynomials useful in numerical integration and asymptotic analysis (Euler–Maclaurin formula [24]); or the Euler polynomials which

ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 206 Sergio A. Carrillo and Miguel Hurtado

lead to Euler–Boole formula [9]. On the other hand, a wide class of Appell sequences provides special values of transcendental functions, as recently proved in [20], extending the well-known case of the Bernoulli polynomials as the values at negative integers of the Hurwitz zeta function. Appell sequences are a subclass of Sheffer sequences (type zero polynomials) which were treated by I. M. Sheffer as solutions of families of differential and difference equations [29]. Later on, their study, leaded by Rota and Roman, developed in what it is known today as Umbral Calculus [26–28]. In contrast, a more recent approach has been made using matrix and determinantal representations, see, e.g., [1,2,13, 14, 37, 38]. Also, current research has focussed on special sequences [16] and other alternative descriptions of the theory, for instance, through random variables [3,33]. Our goal in this work is to obtain a new simple recursion for Appell sequences (Theorem6), their expansions in terms of an arbitrary delta operator (Proposition7) (as presented in [4] for difference operators), and several examples. Our exposition is based on Umbral Calculus, briefly recalled in Sections2 and3. Using this tool, we obtain in Section3 a new recurrence for Sheffer sequences (Theorem5) using left-inverses of a delta operator and deduce Theorem6 as a particular case. In addition, we clarify how the characterization of Sheffer and Appell sequences through a linear functional and a linear operator are naturally equivalent (Theorem3) (a general fact used systematically in specific cases, see, e.g., [9], [16, Theorems 2.2–2.3], [18,34]). Our examples are presented in Section5, in particular, we generalize the Weierstrass operator for classical Hermite polynomials by using Ecalle’s accelerator functions to obtain an integral representation of the re-scaled Hermite d-orthogonal polynomials [15] (Proposition20). We also include the integral representations of the inverse operators associated with the Bernoulli and Euler polynomials (Propositions 13 and 16). These formulas are closely related to their moment expansions [32, 33], but here we deduce them in a direct elementary way. Finally, we collect in Table1 multiple examples of Appell sequences scattered in the literature, with their respective linear functional and characterization.

2. Preliminaries on Sheﬀer sequences

We briefly recall some characterizations of Sheffer sequences and set the notations used along the paper. Our summary is based on the expositions [26, 28] of Umbral Calculus whose cornerstone is the twofold identification of formal power series in one variable C[[t]] as the linear functionals, as well as the shift-invariant linear operators of the ring of univariate polynomials C[x]. Let ∂ ∂ be usual differentiation and T : C[x] C[x], T (p)(x) p(x a) the shift-operator = x a → a = + indexed by a C. A linear operator Q : C[x] C[x] is shift-invariant if Q T T Q, for all a C. ∈ → ◦ a = a ◦ ∈ The set Σ of these operators acquires a commutative ring structure via the isomorphism

X∞ bn n X∞ bn n ι∂ : C[[t]] Σ, B(t) t Q B(∂) ∂ , (1) → = n 0 n! 7−→ = = n 0 n! = = where the product of formal power series corresponds to composition of operators. In fact, bn n n is found by evaluating the polynomial Q(x ) at x 0, i.e., bn Q(x )(0). Now, we can extend P n = P= n Q to C[x][[t]] by the rule P(x,t) ∞ p (x)t Q(P)(x,t) ∞ Q(p )(x)t (we denote this = n 0 n 7→ = n 0 n extension also by Q, and despite the= abuse of notation, its meaning= should be clear from the context). In this way, we can recover B(t) using the value of Q at the exponential: Q(ext )(x,t) xt 1 xt = B(t)e and therefore ι− (Q)(t) Q(e )(0,t), see Remark8 for an example. ∂ = An operator Q Σ is called a delta operator if the polynomial Q(x) is a non-zero constant. In ∈ this case Q(a) 0, for all a C and deg(Q(p)) deg(p) 1. Thus the series B(t) starts at n 1 = ∈ = − P bn n = with b1 Q(x) 0 and it admits a compositional inverse B(t) n∞ 1 n! t . The role of the series = 6= = =

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B(t) is to induce the sequence {qn(x)}n 0 of basic polynomials of Q. These are deﬁned through ≥ the expansion n Ã ! xB(t) X∞ qn(x) n X n e t , and they satisfy qn(x x0) q j (x)qn j (x0). (2) = n 0 n! + = j 0 j − = = Thus, they are of binomial type. Moreover, they are characterized by

q0(x) 1, qn(0) 0, and Q(qn)(x) nqn 1(x), for all n 1. (3) = = = − ≥ The previous properties are easily established. For instance, (2) follows from equating co- (x x0)B(t) xB(t) x0B(t) xB(t) efficients in powers of t in e + e e . Likewise, (1) shows that Q(e )(x,t) ˆ = = P bn n xB(t) xB(t) xB(t) n∞ 1 n! B(t) e B(B(t))e te , equality that proves the last equation in (3). = = = A delta operator Q also induces a ring isomorphism ιQ : C[[t]] Σ by A(t) A(Q), and P an n xB(t)→ 7→xB(t) if S A(Q) n∞ 0 n! Q , then an S(qn)(0). Therefore S(e )(x,t) A(t)e and we = 1 = = xB(t) = = recover ιQ− (S)(t) S(e )(0,t). Also, note that S is invertible if and only if a0 S(1) 0, and 1 = = 6= S− 1/A(Q), where 1/A(t) is the reciprocal of A(t). = We say {sn(x)}n 0 is a Q-Sheffer sequence for the delta operator Q if s0 0 is constant and ≥ 6= Q(sn)(x) nsn 1(x), for all n 1. (4) = − ≥ These sequences admit several characterizations. First, there is an invertible S Σ satisfying ∈ S(s ) q , for all n 0. (5) n = n ≥ Second, its exponential generating series has the form xB(t) X∞ sn(x) e t n , (6) n 0 n! = A(t) = where A(t) C[[t]] has a reciprocal, i.e., A(0) 0. Third, the sequence satisfies ∈ 6= n Ã ! n Ã ! X n X n sn(x x0) sk (x)qn k (x0), and sn(x) sk (0)qn k (x), for all n 0. (7) + = k 0 k − = k 0 k − ≥ = = Conditions (4), (5), (6) and both equations in (7) are equivalent to each other as can be checked. P n The relevant relations are S ιQ(A) and 1/A(t) n∞ 0 sn(0)t /n!. We highlight that S is = = = uniquely associated to {sn(x)}n 0 which will be referred as the Q-Sheffer sequence relative to ≥ S ((Q,S)-Sheffer for short). Finally, after repeated application of Q to (5) followed by evaluating at x 0, we find that = S(Qm(s ))(0) n!δ , for all n,m 0, (8) n = n,m ≥ where δn,m is the Kronecker delta.

3. The use of linear functionals

Another characterization of Sheﬀer sequences is available through functionals of C[x]. It is based on the identiﬁcation of the dual space C[x]∗ with Σ (and thus with C[[t]] via (1)).

Lemma 1. We have the linear isomorphism j : C[x]∗ Σ given by → L L(p)(x): L(T (p)), and having as inverse L L(p) L(p)(0), p C[x]. (9) 7−→ = x 7−→ = ∈ Proof. Given L C[x]∗ the map L j(L) is clearly linear. It is also shift-invariant since ∈ = L(Ta(p))(x) L(Tx (Ta(p))) L(Tx a(p)) L(p)(x a) Ta(L(p))(x), for all a C. = = + = + = ∈ Conversely, if L Σ and L(p) L(p)(0), then L C[x]∗ and L(T (p)) L(T (p))(0) ∈ = ∈ x0 = x0 = (T L)(p)(0) L(p)(x ), for all x C. Thus the maps in (9) are inverses one of each other. x0 ◦ = 0 0 ∈

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n Remark 2. We can codify L C[x]∗ by the values Ln : L(x ) known as the moments of L. P∈ n P n= xB(t) Extending L to C[x][[t]] by L( n∞ 0 pn(x)t ) n∞ 0 L(pn)t we have the relation L(e )(x0,t) xB(t) (x x0)B(t) =x0B(t) xB(t=) = = L(T (e )) L(e + ) e L(e ). In particular, the indicator series of L x0 = = n xt X∞ t xt xt xt L(e ) Ln , satisﬁes L(e )(x,t) L(e )e . (10) = n 0 n! = = Now we are in position to give two equivalent ways to characterize Sheﬀer sequences using functionals.

Theorem 3. Given S C[x]∗, there is a unique Q-Sheffer sequence {sn}n 0 satisfying ∈ ≥ S(s ) 1 and S(s ) 0, for all n 1, (11) 0 = n = ≥ or equivalently, S(T (s )) q (x ), for all n 0 and x C. (12) x0 n = n 0 ≥ 0 ∈ xB(t) xB(t) Indeed, {sn}n 0 is the (Q,S)-Sheffer sequence, S j(S), having generating series e /S(e ). ≥ = Conversely, given a Q-Sheffer sequence {sn}n 0, there is a unique S C[x]∗ such that (11) holds. ≥ ∈ Proof. First note that (11) is simply (12) for x 0 as (3) shows. Conversely, if (11) holds, 0 = then (12) follows by applying S to the first equation in (7). Furthermore, if S j(S), then = equation (5) is equivalent to (12) since S(Tx0 (sn)) S(sn)(x0), thanks to Lemma1. Also, applying S to equation (6) we find A(t) S(exB(t)). = = Finally, given the (Q,S)-Sheffer sequence {sn}n 0, if S0 C[x]∗ satisfies (11), then S0 j(S0) Σ ≥ ∈ = ∈ is invertible and satisfies S0(sn) qn for all n. Since S is characterized by this condition, then 1 = S0 S and S0 j− (S) S is also uniquely determined. = = =

The previous theorem shows that it is equivalent to have a Q-Sheffer sequence {sn}n 0, an ≥ invertible operator S Σ or a functional S C[x]∗ such that S(1) 0. Thus we can refer to {sn}n 0 ∈ ∈ 6= ≥ as the (Q,S,S)-Sheffer sequence, where S j(S). = Remark 4. The k-fold iteration Sk of an invertible operator S Σ produces the (Q,Sk ,Sk )- (k) k 1 k ∈ xB(t) xB(t) k Sheffer sequence {sn }n 0, S : j− (S ), having exponential generating series e /S(e ) . ≥ (k) = ( k) ( k 1) (k 1) According to (5) we find s S◦ − (q ) S S◦ − − (q ) S(s + ) holding for all k,n N. n = n = ◦ n = n ∈ We also highlight that if S admits the representation Z Z S(p) p(s)w(s)ds, and S(p)(x) p(x s)w(s)ds, = I = I + where I R is an interval and w : I C is such that Sn are all finite, then ⊆Z → Z k k S (p) p(s1 sk )w(s1) w(sk )ds, S (p)(x) p(x s1 sk )w(s1) w(sk )ds, = I k + ··· + ··· = I k + + ··· + ··· where ds ds ds and the integration is taken over the kth Cartesian product I k Rk . = 1 ··· k ⊆ A Q-Sheffer sequence can be calculate though its generating series, or using determinants, see [13,14,37]. Here we present a new recursion formula using left-inverses for Q.

1 Theorem 5. Let Q be a delta operator and B ι− (Q). Then each x C deﬁnes = ∂ 0 ∈ ∂ µZ x ¶ Q 1 Q −x0 (p): p(s)ds , which is a left-inverse for . = B(∂) x0

Here the integral is taken over the line segment from x0 to x. Through it, the (Q,S,S)-Sheﬀer sequence {sn}n 0 can be calculated recursively by s0 1/S(1) and ≥ = n Q 1 ¡Q 1 ¢ sn n −x0 (sn 1) S −x0 (sn 1) , n 1. (13) = − − s0 − ≥

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1 Proof. By writing Q ∂ P, where P Σ is invertible and P− ∂/B(∂), we ﬁnd = ◦ ∈ = µ µZ x ¶¶ µZ x ¶ Q Q 1 P P 1 −x0 (p)(x) (∂ ) − p(s)ds ∂ p(s)ds p(x), ◦ = ◦ x0 = x0 = as required. Now, setting s0 s 1/S(1) and s0 (x) for the left-side of (13) we have 0 = 0 = n n Q Q Q 1 Q 1 ¡Q 1 ¢ (sn0 ) n ( −x0 (sn0 1)) nsn0 1, and S(sn0 ) nS( −x0 (sn0 1)) S −x0 (sn0 1) S(1) 0, = − = − = − − s0 − = for all n 1. Thus {sn0 }n 0 is the (Q,j(S),S)-sequence, so sn sn0 , for all n. ≥ ≥ =

4. The case of Appell sequences

The main example of Sheﬀer sequences are the Appell sequences (in honor of P.E. Appell (1880) [6]) corresponding to Q ∂, for which B(t) B(t) t, and q (x) xn. (14) = = = n = In this case we see {pn(x)}n 0 is an Appell sequence if p0 0 is a constant and ≥ 6= dpn (x) npn 1(x), n 1, (15) dx = − ≥ n Equivalently, there is a unique invertible L Σ satisfying L(pn) x , for all n 0, the sequence xt ∈ xt 1= ≥ has a exponential series of the form e /L(e ), where L j− (L) C[x]∗, or they satisfy the = ∈ corresponding equations to (7). We can refer to {pn}n 0 as the (L,L)-Appell sequence. In this ≥ setting, Theorems3 and5 take the following form.

Theorem 6. The (L,L)-Appell sequence {pn}n 0 is the unique Appell sequence satisfying ≥ L(T (p )) xn, for all n 0 and x C. (16) x0 n = 0 ≥ 0 ∈ Furthermore, it can be calculated recursively by p 1/L(1) and 0 = Z x n µZ x ¶ pn(x) n pn 1(s)ds L pn 1(s)ds . (17) = x0 − − p0 x0 − Although we have deduced the previous theorem from the more general case of Sheﬀer sequences, it can obtained by directly means. We also remark that (16) is referred as the mean value property for Appell sequences connected to random variables, see [33, Proposition 2.7] and the references therein. Additionally, we can also express an Appell sequence in terms of a delta operator as follows.

xt xt Proposition 7. Let {pn}n 0 be the (L,L)-Appell sequence with generating series e /L(e ) xt ≥ P αk k= C(t)e . If Q is a delta operator with B(t) C[[t]] as in equation (2), and (C B)(t) ∞ t , ∈ ◦ = k 0 k! then = n 1 X∞ αk k 1 n X αk k n L− Q , and thus pn(x) L− (x ) Q (x ). = k 0 k! = = k 0 k! = = P αk k xt Proof. The operator Q ∞ Q Σ is invertible since α C(0) 0. Recalling that Q(e ) 1 = k 0 k! ∈ 0 = 6= = B(t)ext , we ﬁnd = n X∞ n t xt xt xt X∞ pn(x) n Q1(x ) Q1(e ) (C B)(B(t))e C(t)e t . n 0 n! = = ◦ = = n 0 n! = = 1 n n 1 n Therefore, L− (x ) pn(x) Q1(x ), for all n, and Q1 L− as required. Finally, Q1(x ) Pn αk k n = k n = k = = k 0 k! Q (x ), since Q (x ) 0 if k n as Q lowers the degree of a polynomial by k. = = >

C. R. Mathématique, 2021, 359, n 2, 205-217 210 Sergio A. Carrillo and Miguel Hurtado

Remark 8. The previous proposition was recently studied in [4] for Q ∆ ∆, the difference = 1 = operator of step one. Let us recall that for each h C∗, the difference operator ∈ ∆ : T 1, i.e., ∆ (p)(x) p(x h) p(x), h = h − h = + − xt xt ht ht constitute a delta operator for which B(t) ∆h(e )(0,t) e (e 1) x 0 e 1 and B(t) xB(t) x/h P = n = − | = = − = log(1 t)/h. Thus e (1 t) ∞ (x/h) t /n! and ∆ has q (x) (x/h) as basic + = + = n 0 n h n = n sequence. Here (a) : a(a 1) (a n 1)= is the falling factorial. n = − ··· − + Remark 9. Any pair of series B(t),B(t) tC[[t]], B 0(0) 0, B 0(0) 0, compositional inverses one ∈ 6= 6= of each other, define two families of numbers {sB (n,k)}n k and {SB (n,k)}n k determined by ≥ ≥ k n k n B(t) X∞ t B(t) X∞ t sB (n,k) , SB (n,k) , (18) k! = n k n! k! = n k n! = = just as B(t) log(1 t), B(t) et 1 define the Stirling numbers of first and second kind [12, p. 50], = + = − k n t k n log(1 t) X∞ t (e 1) X∞ t + s(n,k) , − S(n,k) . (19) k! = n k n! k! = n k n! = = If B,B are associated to the delta operator Q as in (2), then (18) induces the inverse relations n n X k n X qn(x) sB (n,k)x , x SB (n,k)qk (x), = k 0 = k 0 = = and n n X X αk sB (n,k)pk (0), pk (0) SB (n,k)αk , = k 0 = k 0 = = xB(t) xt where {αk }k 0 and {pk (0)}k 0 are as in Proposition7. These follows from expanding e , e ¯ ≥ ≥ = exB(B(t)), C(t), and C(t) (C B)(B(t)), respectively, as in the usual case of Stirling numbers [12, = ◦ p. 144].

5. Examples

This ﬁnal section is aimed to apply the previous results to concrete examples focusing on the role of the functional involved. In particular, we ﬁnd integral representations for Bernoulli and Euler polynomials, and also for Hermite d-orthogonal polynomials [15]. Finally, we collect in Table1a list of important (L,L)-Appell sequences and their characterization via Theorem6, including the recent results of Kummer hypergeometric polynomials given in [16].

Remark 10. Any functional S C[x]∗ admits a representation of the form Z ∈ Z S(p) +∞ p(s)dβ(s), S(p)(x) +∞ p(x s)dβ(s), = 0 = 0 + for some function β : (0, ) C of bounded variation as it was proved by Boas [8] in relation to +∞ → the Stieljes moment problem. Thus the characterization of Theorem3 can be written as Z +∞ sn(x0 s)dβ(s) qn(x0), 0 + = R (m) and equation (8) takes the form +∞ Q (s )(s)dβ(s) n!δ . This reasoning contains the early 0 n = n,m characterization of Appell sequences of Thorne [35], soon after generalized by Sheﬀer [30].

xt Example 11. Let {pn(x)}n 0 be the (L,L)-Appell sequence, C(t) 1/L(e ), and α,β C with ≥ = n ∈ β 0. Then {pn(x α)}n 0 is the (L Tα,L Tα)-Appell sequence and {β− pn(βx)}n 0 is the 6= − ≥ ◦ ◦ ≥ (L Hα,L Hα)-Appell sequences, where Hβ : C[x] C[x] is the homothecy Hβ(p)(x) p(x/β). ◦ ◦ → (x α)t xt (x α)t = xt Indeed, these sequences have as generating series C(t)e − e /L(e + ), and C(t/β)e = = ext /L(ext/β), respectively.

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Example 12 (Bernoulli polynomials). They are deﬁned by the expansion

n n Ã ! X∞ t t xt X n n j Bn(x) t e , Bn(x) B j (0)x − . n 0 n! = e 1 = j 0 j = − = 1 The B j B j (0) are the Bernoulli numbers that satisfy B0 1, B1 and B2j 1 0, j 1. We see = = = − 2 + = ≥ the Bernoulli polynomials conform the Appell sequence relative to Z 1 Z 1 et 1 I(p) p(s)ds and I(p)(x): p(x s)ds, since I(ext ) − . = 0 = 0 + = t R 1 n Theorem6 asserts they are characterized by the condition I(Bn)(x) 0 Bn(x s)ds x , or = + n 1 = equivalently after differentiation, by the equation ∆(B )(x) B (x 1) B (x) nx − , which is n = n + − n = a well-known result. Furthermore, we can compute them recursively by B (x) 1 and 0 = Z x Z 1 Z u Bn(x) n Bn 1(t)dt n Bn 1(t)dtdu. = 0 − − 0 0 − k R Following Remark4 we find the k-fold iteration of I is I (p) k p(s s )ds which pro- = [0,1] 1+···+ k duces the kth order Bernoulli polynomials B (k)(x) having t k ext /(et 1)k as generating exponen- n − tial series. These are also known as Nørlund polynomials due to N. E. Nørlund who introduced them in 1922 [22]. Moreover, we can write these polynomials in terms of ∆ as n j n X ( 1) j n (k) X k! j n Bn(x) − ∆ (x ), and Bn (x) s(k j,k)∆ (x ), = j 0 j 1 = j 0 (k j)! + = + = + by using Proposition7. In fact, in this case B(t) log(1 t) and the previous formulas follow = + from (19) since ((et 1)/t)k B(t) log(1 t)k /t k , c.f., [4, Theorem 5]. − ◦ = + An interesting question is to determine an analytic representation for the inverse operator of I which in turn gives left-inverses for ∆ and an analytic representation of Bernoulli polynomials. We remark that the formula below to invert ∆ is familiar in the theory of difference equations and it has been used to justify Ramanujan summation, see [11, Theorem 1].

p0(0) R p0(i s) p0( i s) xt t Proposition 13. The map L(p) p(0) i +∞ − − ds verifies L(e ) t/(e 1). = − 2 − 0 e2πs 1 = − Therefore, the inverse operator of I(p)(x) R 1 p(x s)ds is − = 0 + Z p0(x) p0(x i s) p0(x i s) I 1(p)(x) p(x) i +∞ + − − ds. − 2πs = − 2 − 0 e 1 − Moreover, the difference operator ∆ admits the left-inverses Z x p(x) Z p(x i s) p(x i s) ∆ 1(p) p(s)ds i +∞ + − − ds. −x0 2πs = x − 2 − 0 e 1 0 − Furthermore, the Bernoulli polynomials admit the integral representation Z n 1 n 1 n (x i s) − (x i s) − B (x) xn xn 1 in +∞ + − − ds. n − 2πs = − 2 − 0 e 1 − 2j 1 Proof. For the first statement note L(1) 1 B0, L(x) 1/2 B1 and L(x + ) 0, j 1 since 2j 1 = = = − = = ≥ the derivative of x + is an even function. For the even powers we find Z 2j 1 2s − L(x2j ) 2j( 1)j 1 +∞ ds B , + 2πs 2j = − 0 e 1 = − values that are familiar in the study of Abel–Plana formula [24, p. 291], [25, 24.7.2]. Now, the xt t 1 operator L j(L) is the inverse of I since I(e ) (e 1)/t. Finally, Proposition5 shows ∆− (p) = = − x0 = ∂ ¡R x p(s)ds¢ and the previous example proves B (x) I 1(xn) as required. eh∂ 1 x0 n − − =

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Example 14 (Euler polynomials). We consider the Apostol–Euler polynomials determined by xt n e X∞ t t En(β;x) , for a ﬁxed β 0. (20) 1 β(e 1) = n 0 n! 6= + − = They are the Appell sequence relative to

L(p) (1 β)p(0) βp(1), L(p)(x) (1 β)p(x) βp(x 1), since L(ext ) 1 β(et 1). = − + = − + + = + − The case β 1/2 recovers the classical Euler polynomials E (1/2;x) E (x). Theorem6 shows the = n = n Apostol–Euler are characterized by (1 β)E (β;x) βE (β;x 1) xn. Moreover, they are given − n + n + = recursively by E (β;x) 1 and 0 = Z x Z 1 Z u En(β;x) n En 1(β;t)dt βn En 1(β;t)dtdu. = 0 − − 0 0 − (k) k The kth Apostol–Euler polynomials En (β;x) are the Appell sequence relative to L (p) Pk ¡k¢ j k j xt ¡ t ¢k = j 0 j β (1 β) − p(j) and with exponential generating series e / 1 β(e 1) . The case = − + − β 2 corresponds to the kth Euler polynomials E (k)(x). Finally, Proposition7 proves that = n n Ã ! n j X j k 1 X ( 1) E (k)(β;x) + − ( 1)j βj ∆j (xn), and E (x) − ∆j (xn), n n j = j 0 j − = j 0 2 = = t k k P ¡j k 1¢ j j j since (1 β(e 1))− log(1 t) (1 βt)− ∞j 0 + j − ( 1) β t , c.f., [4, Theorem 7]. + − ◦ + = + = = − Remark 15. It is worth recalling that Apostol–Euler polynomials are usually deﬁned through the xt t expansion 2e /(λe 1), for a parameter λ C∗, which up to a constant is equivalent to (20). + ∈ On the other hand, these can be expressed in terms of Apostol–Bernoulli polynomials that were introduced by T. Apostol in relation with the Lerch zeta function [5], see, e.g., [21] for more details on these connections.

In analogy with Proposition 13, we can write an analytic expression for the inverse of the operator inducing the Euler polynomials. More specifically, we have. ¡ ¢ ¡ ¢ p 1 i s p 1 i s R − 2 + 2 + − 2 − 2 xt t Proposition 16. The functional L(p) 0+∞ πs/2 πs/2 ds satisfies L(e ) 2/(e 1). In = e e− = + consequence, + Z p ¡x 1 i s ¢ p ¡x 1 i s ¢ J 1(p)(x) +∞ − 2 + 2 + − 2 − 2 ds − πs/2 πs/2 = 0 e e + − is the inverse of J(p)(x) (p(x 1) p(x))/2. Furthermore, the Euler polynomials admit the integral = + + representation n n Z ¡x 1 i s ¢ ¡x 1 i s ¢ E (x) +∞ − 2 + 2 + − 2 − 2 ds. n πs/2 πs/2 = 0 e e + − Proof. It is sufficient to establish the first formula, the remaining ones follow as in the previous proposition. For this purpose we write the Euler polynomials in terms of the Euler numbers En as n Ã ! k µ ¶n k n X n i 1 − 2 X∞ t E (x) E x where i nE . n k k t t n = k 0 k 2 − 2 e e− = n 0 n! = + = P t n xt t (x 1/2)t t/2 t/2 In fact, n∞ 0 En(x) 2e /(e 1) 2e − /(e e− ). Also note that E2j 1 0, j 0. Now, n! = + = + + = ≥ by Example= 11, it is enough to show that the operator

Z p ¡ i s ¢ p ¡ i s ¢ L (p) +∞ 2 + − 2 ds, 0 πs/2 πs/2 = 0 e e + −

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xt t/2 t/2 xt t/2 xt satisfies L0(e ) 2/(e e− ). In fact, L L0 T 1/2 and therefore L(e ) e− L0(e ) t = + 2j 1= ◦ − = = 2/(e 1) as required. It is clear that L0(x + ) 0 since these are odd functions. For the even + = powers we also find [25, 24.7.6] j 2j ( 1) Z s E2j L (x2j ) − +∞ ds ( 1)j , 0 2j 1 πs/2 πs/2 2j = 2 0 e e = − 2 − + − as required. Remark 17. Although the integral representations of Euler numbers 2j µ ¶2j 1 2j Z s 2 + Z ln(u) E 2 +∞ ds +∞ du, 2j πs/2 πs/2 2 = 0 e e = π 0 u 1 + − + are known (here u eπs/2), we find instructive to include a simple proof using calculus of = n 1 R ln(u)n residues. Indeed, we can find A (2/π) + +∞ du, recursively: using the branch of the n = 0 1 u2 logarithm log(z) ln z iarg(z) with π/2 arg(z+) 3π/2, the Residue Theorem shows that n n R log(z) = | ¡|log( + z) ¢ n n− 1 n< < 2 dz 2πi Res 2 ,i i π + /2 . Here 0 ² 1, R 1 and γ²,R is the path formed γ²,R z 1 = z 1 = < < > by the segments+ from R to+ ² and ² to R, and the corresponding semicircles centered at 0 of − − radius ² and R, oriented positively. Letting ² 0+ and R the integral over the arcs tends to → → +∞ 0 and we obtain Z n Z n n n 1 ln(u) (ln(u) iπ) i π + +∞ +∞ + 2 du 2 du n . 0 1 u + 0 1 u = 2 + + Pn 1 ¡n¢ k k n P n Thus the sequence An satisfies An k−0 k i 2 An k 2i . If we set A(t) n∞ 0 An t /n!, this + = 2i t− = i t = = i t i t recursion is equivalent to the equation A(t) A(t)e 2e . Consequently, A(t) 2/(e e− ) P n n n + = = + = n∞ 0( 1) En t /n! and An ( 1) En as needed. = − = − Now we proceed to extend the integral representation and characterization of Hermite polynomials that are essentially the only Appell orthogonal sequence [31]. But first we need a remark.

Remark 18. Given L C[x]∗ and an integer m 1, we can construct a functional recording only ∈ ≥ the moments of L indexed by multiples of m. Indeed, recalling Example 11 and fixing the m-th root of unity ω : e2πi/m, we see that the functional m = 1 Lm (L L Hω 1 L H (m 1) ) = m + ◦ m− + ··· + ◦ ωm− − nm xt P t nm has moments Lm(x ) Lnm and equal to zero otherwise, i.e., Lm(e ) n∞ 0 Lnm (nm)! . This = j j(m 1) = = can be checked using the identity 1 ω ω − 0, valid for j 1,...,m 1. + m + ··· + m = = − Example 19 (Hermite polynomials). Fix an integer d 1. We shall describe an analytic expres- ≥ sion for the functional defining the Appell sequence determined by the expansion n ³ d 1´ X∞ (d) t exp xt t + Hn (x) , − = n 0 n! = which correspond to a particular case of Gould–Hopper polynomials [17]. Our approach is based on Ecalle’saccelerator operators familiar in the theory of multisummability of power series, see [7, Chapter 11]. To this end, we recall the accelerator function µ ¶ µ ¶ n 1 X∞ (n 1)π n 1 z 1 1 Cα(z): sin + Γ + , where α 1, 1, (21) = π n 0 β α n! > α + β = = and Γ is the Gamma function. The map C is entire and satisfies C (z) c exp( c z β) on α | α | ≤ 1 − 2| | each sector arg(z) θ/2 π/(2β), for certain constants c j c j (α,θ) 0, j 1,2. Then, given | | ≤ < = k R> = k k k0 k 0, the acceleator operator of index (k0,k), A (p)(z): z− +∞ p(s)C ((s/z) )ds > > k0,k = 0 k0/k

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C n is well-defined for all polynomials p [z]. Its importance relies on the fact that Ak0,k (z ) Γ(1 n/k) n ∈ = Γ(1 +n/k ) z , for all n 0. Choosing k 1, k0 d 1, and z 1, we find that + 0 ≥ = = + = Z +∞ n n! Ad 1,1(p): p(s)Cd 1(s)ds has moments Ad 1,1(x ) . + = + + = ¡ n ¢ 0 Γ 1 d 1 + + 2πi/(d 1) Therefore, if ωd 1 e + , Remark18 proves that the functional + = Z d 1 1 +∞ ³ d ´ A + (p): p(s) p(ωd 1s) p(ωd 1s) Cd 1(s)ds (22) = d 1 0 + + + ··· + + + + d 1 xt d 1 d 1 satisfies A + (e ) exp(t + ). Moreover, for λ 0, Example 11 shows that A + Hλ produces n =(d) 6= d 1 ◦ the sequence {λ− Hn (λx)}n 0 having exp(xt (t/λ) + ) as generating series. In particular, if d 1 ≥ iπ/(d 1) − d 1 we choose λ + 1, say λ e + , the generating series would be exp(xt t + ). These = − = + considerations establish the following. (d) d 1 Proposition 20. The sequence {Hn (x)}n 0 having exp(xt t + ) as exponential generating series ≥ d 1 − is characterized as the Appell sequence relative to A + given by (22) and satisfying Z d +∞ ³ 2πi j ´ X (d) d 1 n Hn x e + s Cd 1(s)ds (d 1)x , (23) 0 j 0 + + = + = (d) where Cd 1 is Ecalle’s accelerator function (21). Moreover, Hn admits the integral representation + Z d n 1 +∞ ³ 2πi(j 1/2) ´ (d) X d −1 Hn (x) x e + s Cd 1(s)ds. (24) = d 1 0 j 0 + + + = 2 1/(d 1) On the other hand, if in the previous paragraph we take λ λd : (d!(d 1) ) + we recover n (d) = = + the family Hbn(x;d): λ− H (λd x) of Hermite-type-d-orthogonal polynomials. We briefly recall = d n that a sequence of polynomials {pn(x)}n 0 with deg(pn) n, is d-orthogonal with respect to a d ≥ = functionals L ,...,L C[x]∗ if 1 d ∈ L j (pm pn) 0, for n md j, and L j (pm pmd j 1) 0, for all m 0, j 1,...,d, = ≥ + + − 6= ≥ = see [15, Definition 1.5] and the references therein. For d 1 this corresponds to the classical or- = thogonality condition. Moreover, d-orthogonal Appell polynomial sequences have a generating Pd 1 j 2 series of the form exp(xt − γd 1 j t + /(j 2)!), for certain constants γl [15, Theorem 3.1]. − j 0 − − + Finally, in the case d 1 and= α β 2 in (21), after a direct calculation using the values = = = Γ(n 1/2) (2n)!/(4nn!)pπ, n 0, we find the familiar function C (z) exp( z2/4)/pπ. Then, + = ≥ 2 = − taking λ p2 we find the functional = 1 Z ³ ´ 1 Z 2 2 +∞ p p +∞ s /2 (A Hp2)(p) p(s/ 2) p( s/ 2) C2(s)ds p(s)e− ds, ◦ = 2 0 + − = p2π −∞ corresponding to the Weierstrass operator [16, p. 746] who induces the classical Hermite polyno- 2 mials {Hen(x)}n 0 with generating series exp(xt t /2). Finally, equations (23) and (24) take the ≥ − familiar form [19, p. 254] Z Z n 1 +∞ s2/2 1 +∞ n s2/2 x Hen(x s)e− ds, Hen(x) (x i s) e− ds. = p2π + = p2π + −∞ −∞ We conclude with one final worked example in relation with functionals induced by entire functions.

P n Example 21. Given an entire function F (z) n∞ 0 fn z and λ C such that F (λ) 0, let = = ∈ 6= 1 X∞ k 1 X∞ k LF,λ(p) p(k)fk λ , and LF,λ(p)(x) p(x k)fk λ . = F (λ) k 0 = F (λ) k 0 + = =

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∂ j To see these are well-deﬁned we use the diﬀerential operator δ z ∂z to note that δ (F )(z) P j k = = k∞ 1 k fk z , j 1 are again entire and thus = ≥ n n Ã ! j n δ (F )(λ) n X n δ (F )(λ) n j LF,λ(x ) , and LF,λ(x )(x) x − . = F (λ) = j 0 j F (λ) = xt P kt k t Moreover, LF,λ(e ) k∞ 0 e fk λ /F (λ) F (λe )/F (λ). In this way we obtain an Appell se- = = = quence {PF,λ,n(x)}n 0 characterized by the equation ≥ X∞ k n PF,λ,n(x k)fk λ F (λ)x . k 0 + = = P k xt If F has no zeros in the complex plane, 1/F (z) k∞ 0 fk0 z is again entire and L1/F,λ(e ) t 1 = = = F (λ)/F (λe ). Therefore, L− L and we can invert the previous equation to obtain F,λ = 1/F,λ n Ã ! j X∞ n k X n δ (1/F )(λ) n j PF,λ,n(x) F (λ) (x k) fk0 λ x − . (25) = k 0 + = j 0 j (1/F )(λ) = = Examples of this situation are given by F (z) exp(P(z)), where P is a non-constant polynomial = as considered by Touchard [36]. The best-known case corresponds to F (z) ez , for which = k n k xt t X∞ t λ n z X λ Lez ,λ(e ) exp(λ(e 1)) Tk (λ) , where Tk (λ) e− δ (e )(λ) S(n,k) = − = k 0 k! = · = k 0 k! = = are the exponential or Touchard polynomials [26, p. 63] (recall equation (19)). In particular, (25) n ¡n¢ j l n j z P P takes the form Pe ,λ,n(x) j 0 j l 0 S(j,l)( λ) x − /l!, see [9, Example 4.4]. = = = − Table1 contains more examples of Appell sequences as Nørlund [23], Laguerre [26, p. 108], Strodt [9], and the Bernoulli-type polynomials [34] (see Lemma 1 where a should be only equal to 1). Moreover, the Bernoulli hypergeometric [18] and Kummer hypergeometric Bernoulli polynomials [16].

Table 1. Some families of Appell polynomials

Polynomials Functional L(p) Indicator series L(ext ) / Characterization Monomials at (x a)n p(a) e − t Bernoulli Z 1 (e 1)/t p(t)dt − Bn(x) 0 n Bn 1(x 1) Bn 1(x) (n 1)x + + − + = + Z (et 1)k /t k kth Bernoulli − (k) p(s1 sk )ds Z Bn (x) [0,1]k + ··· + (k) n Bn (x s1 sk )ds x [0,1]k + + ··· + = k ω t Bernoulli Z Y e j 1 p(ω1s1 ωk sk )ds − k Nørlund [0,1] + ··· + j 1 ωj t = Z k (k) k (k) X n Bn (x ω) ω (ω1,...,ωk ) C Bn (x ωj s j ω)ds x k | = ∈ [0,1] + j 1 | = = Apostol 1 β(et 1) Euler (1 β)p(0) βp(1) + − − + n En(β;x) (1 β)E (β;x) βE (β;x 1) x − n + n + =

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t Euler p(0) p(1) (1 e )/2 + + En(x) 2 E (x) E (x 1) 2xn n + n + = kth Apostol Ã ! ¡ t ¢k k k 1 β(e 1) Euler X j k j + − β (1 β) − p(j) Ã ! j 0 j − k k (k) = X j k j (k) n En (β;x) β (1 β) − En (β;x j) x j 0 j − + = = µ k ¶ k Euler k X X k Y ω t 2− p n ω 2− (e j 1) Nørlund j j + n j {0,1} j 1 j 1 ∈ = = Ã k ! X X E (k)(x ω) ω (ω ,...,ω ) Ck E (k) x n ω ω 2k xn n | = 1 k ∈ n + j j | = n j {0,1} j 1 ∈ = N N X ¡ ¢ X x j t w-Strodt w j p x j w j e j 1 j 1 Sn,w (x) = N n 1 = X X− ¡ ¢ n x j R, 0 w j 1, w j 1 Sn,w x w j x ∈ < < j 1 = j 0 + = = = m Z j m X X j t Bernoulli-type a j p(s)ds a j e /t j l 0 j l = m m m = (m l) X X X (m l) n 1 Bn,1− (x) l,m Z, a j 0, j a j 1 a j Bn,1− (x j) nx − ∈ j l = j l = j l + = = = = Hermite Z exp(t 2/2) 1 +∞ s2/2 p(s)e− ds Z p2π +∞ s2/2 n He (x) −∞ He (x s)e− ds p2πx n n + = −∞ d 1 d-Hermite Z d 2πi j exp(t ), d 0 integer +∞ X ³ ´ Cd 1(s) + p e d 1 s + ds ≥ + Z d 0 j 0 d 1 +∞ ³ 2πi j ´ Cd 1 (d) X (d) d 1 n Hn (x) = + Hn x e + s + (s)ds x 0 j 0 + d 1 = = + α 1 Laguerre Z α s (1 t)− − +∞ s e− − p(s) ds, Re(α) 1 Z α t n n n (α n) 0 Γ(1 α) > − +∞ (α n) t e− ( 1) x ( 1) n!Ln − (x) + Ln − (x t) dt − − 0 + Γ(1 α) = n! Ã N 1 +! Bernoulli t X− j N Z 1 e t /j! /(t /N!) hypergeometric N 1 − N p(s)(1 s) − ds, N 1 j 0 0 − ≥ Z 1 = N 1 n BN,n(x) N BN,n(x s)(1 s) − ds x 0 + − = Z 1 n Kummer Γ(a b) a 1 b 1 X∞ a(a 1) (a n 1) t + p(s)s − (1 s) − ds 1 + ··· + − hypergeometric Γ(a)Γ(b) 0 − + n 1 (a b)(a b 1) (a b n 1) n! Z 1 = + +a 1+ ···b 1+ + − n s − (1 s) − x Ba,b,n(x) Re(a),Re(b) 0 Ba,b,n(x s) − ds > 0 + Γ(a) Γ(b) = Γ(a b) + 1 X∞ p(x k) + F (k)(0)λk F (λet )/F (λ) F (λ) k! PF,λ,n(x) k 0 = (k) k X∞ F (0)λ n F : C C entire, F (λ) 0 PF,λ,n(x k) F (λ)x → 6= k 0 + k! = = Acknowledgments We want to thank the referee for their careful reading of the manuscript and suggestions to improve it.

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