Abstract We revisit the theory of Sheffer sequences by means of the formalism introducedin Rota and Taylor (SIAM J Math Anal 25(2):694–711, 1994) and developed in Di Nardo andSenato (Umbral nature of the Poisson random variables. Algebraic combinatorics and com-puter science, pp 245–256, Springer Italia, Milan, 2001, European J Combin 27(3):394–413,2006). The advantage of this approach is twofold. First, this new syntax allows us noteworthycomputational simplification and conceptual clarification in several topics involving Sheffersequences, most of the open questions proposed in Taylor (Comput Math Appl 41:1085–1098, 2001) finds answer. Second, most of the results presented can be easily implementedin a symbolic language. To get a general idea of the effectiveness of this symbolic approach,we provide a formula linking connection constants and Riordan arrays via generalized Bellpolynomials, here defined. Moreover, this link allows us to smooth out many results involvingBell Polynomials and Lagrange inversion formula.
E. Di Nardo · D. Senato (B)Dipartimento di Matematica e Informatica, Università degli Studi della Basilicata,Viale dell’Ateneo Lucano 10, 85100 Potenza, Italiae-mail: [email protected]
H. NiederhausenDepartment of Mathematical Sciences, Florida Atlantic University,777 Glades Road, Boca Raton, FL 33431-0991, USAe-mail: [email protected]
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1 Introduction
Sheffer polynomials are a large class of polynomial sequences that include Laguerre polyno-mials, first and second kind Meixner polynomials, Bernoulli polynomials, Poisson–Charlierpolynomials, Stirling polynomials and many others. The subject traces back to the end ofthirties [24], but it is still a relevant topic because of its applications in dealing with a varietyof subjects. Sheffer sequences of integral type [3] have applications in signal processing[2,16]. More recently, a connection between Sheffer polynomials and Lévy processes hasbeen investigated in Ref. [17], see references therein. Applications in financial mathematicscan be found in Ref. [23], and applications in statistics can be found in Ref. [15]. Symboliccomputations of classical, boolean and free cumulants using generalized Abel polynomials(a subclass of Sheffer polynomials) have been given in Ref. [11]. New applications of Shefferpolynomials have also been found in mathematical physics, as reported in Ref. [29].
One of the most successful language commonly used for handling Sheffer sequences is theso-called umbral calculus. As well known, umbral calculus was formalized in the language oflinear operators by Gian-Carlo Rota in a series of papers that have produced a plenty of appli-cations (see references in Ref. [7]). In 1994, Rota and Taylor [20] came back to the foundationof umbral calculus with the aim to restore, in a light formal setting, the computational powerof the original tools, heuristically applied by founders Blissard, Cayley and Sylvester. In thisnew setting, to which we refer as the classical umbral calculus, there are two basic devices.The first one is to represent a unital sequence of numbers by a symbol α, called an umbra,that is, to represent the sequence 1, a1, a2, . . . by means of the sequence 1, α, α2, . . . of pow-ers of α via an operator E, resembling the expectation operator of random variables. Thesecond device is to represent the same sequence 1, a1, a2, . . . by means of distinct umbrae,as it happens also in probability theory with independent and identically distributed randomvariables.
At first glance, the classical umbral calculus seems just a notation for dealing with expo-nential generating functions. Nevertheless, this new syntax has given rise noteworthy com-putational simplifications and conceptual clarifications in different contexts. Applications tobilinear generating functions for polynomial sequences are given by Gessel [13]. Connec-tions with wavelet theory have been investigated in Refs. [22,25]. In Ref. [10], the theoryof k-statistics and polykays has been completely rewritten, carrying out a unifying frame-work for these estimators, in both the univariate and multivariate cases. Moreover, very fastalgorithms for computing these estimators have been carried out.
Apart from the preliminary paper of Taylor [28], Sheffer sequences have not been describedin terms of umbrae. Indeed, most of the recent papers involving Sheffer sequences still usethe language of linear operators (see for instance [31]). We believe that a symbolic theoryof Sheffer sequences in terms of umbrae has a number of advantages. Here, by keeping thelength of the present paper within bounds, we have chosen two fundamental topics, pre-sented in Sect. 5, to which this new theory of Sheffer sequences can be fruitfully applied.The first application shows the connection between Abel and binomial polynomials. Thisconnection allows us to simplify many results involving Bell polynomials and the Lagrangeinversion formula. Moreover, this relation has smoothed the way to a symbolic treatment offree cumulant theory [12]. The latter application gives a closed form formula expressing thecoefficients of a polynomial sequence {sn(x)} in terms of a different sequence of polynomials{pn(x)}, which is particularly suited to be implemented in symbolic software. It is provedthat this closed form formula gives also the expression of the elements of Riordan arrays.
The rest of the paper is structured as follows. In Sect. 2, we introduce terminology, nota-tion and some basic definitions of the umbral calculus, adding the notion of the adjoint of an
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umbra. Using umbral polynomials, we stress a feature of the classical umbral calculus, thatis the construction of new umbrae by suitable symbolic substitutions. Section 3 is devotedto Sheffer umbrae and their characterizations. In particular, Theorem 3.1 gives the umbralversion of the well-known Sheffer identity with respect to the associated sequence. Theo-rem 3.3 gives a second characterization, that is {sn(x)} is said to be a Sheffer sequence withrespect to a delta operator Q, when Qsn(x) = nsn−1(x) for all n ≥ 0. Finally, in Sect. 4, wecharacterize two special Sheffer umbrae whose moments are binomial and Appell sequences,respectively.
2 The classical umbral calculus
In the following, we recall terminology, notation and some basic definitions of the classi-cal umbral calculus, as it has been introduced by Rota and Taylor in Ref. [20] and furtherdeveloped in Refs. [8,9]. An umbral calculus consists of the following data:
a) a set A = {α, β, . . .}, called the alphabet, whose elements are named umbrae;b) a commutative integral domain R whose quotient field is of characteristic zero;c) a linear functional E, called evaluation, defined on the polynomial ring R[A] and taking
values in R, such that E[1] = 1 and
E[αiβ j · · · γ k
]= E
[αi
]E
[β j
]· · · E
[γ k
], (uncorrelation property)
for any set of distinct umbrae in A and for i, j, . . . , k nonnegative integers.d) an element ε ∈ A, called augmentation [19], such that E[εn] = δ0,n, for any nonnegative
integer n, and an element u ∈ A, called unity umbra [8], such that E[un] = 1, for anynonnegative integer n.
A sequence a0 = 1, a1, a2, . . . in R is umbrally represented by an umbra α when E[αn] =an for all nonnegative integers n. The elements {an} are called moments of the umbra α.
Example 2.1 Singleton umbra. The singleton umbra χ is such that E[χ1] = 1 and E[χn] = 0for n = 2, 3, . . ..
The factorial moments of an umbra α are the elements a(0) = 1, a(n) = E[(α)n] for allnonnegative integers n, where (α)n = α(α − 1) · · · (α − n + 1) is the lower factorial.
Example 2.2 Bell umbra. The Bell umbra β is such that E[(β)n] = 1 for all nonnegativeintegers n. The moments of the Bell umbra are the Bell numbers {Bn}, E[βn] = Bn, i.e. thenumber of partitions of a finite nonempty set with n elements, or the n-th coefficient in theTaylor series expansion of the function exp(et − 1).
An umbral polynomial is a polynomial p ∈ R[A]. The support of p is the set of all umbraeoccurring in p. If p and q are two umbral polynomials, then p and q are uncorrelated if andonly if their supports are disjoint. Moreover, p and q are umbrally equivalent if and only ifE[p] = E[q], in symbols p � q .
Similar umbrae and dot-product Two umbrae α and γ are similar when αn is umbrallyequivalent to γ n, for all n = 0, 1, 2, . . . in symbols α ≡ γ ⇔ αn � γ n .
Example 2.3 Bernoulli umbra. The Bernoulli umbra ι is characterized by the followingumbral relations ι + u ≡ −ι and (ι + u)n � ιn for n > 1. Its moments are the Bernoullinumbers [20].
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Thanks to the notion of similar umbrae, the alphabet A has been extended with theso-called auxiliary umbrae, resulting from operations among similar umbrae. This leadsto the construction of a saturated umbral calculus, in which auxiliary umbrae are handled aselements of the alphabet. The symbol n.α denotes the dot-product of n and α, an auxiliaryumbra [20] similar to the sum α′ + α′′ + · · · + α′′′, where {α′, α′′, . . . , α′′′} is a set of ndistinct umbrae, each one similar to the umbra α. We assume 0.α ≡ ε. Properties of thedot-product have been investigated with full details in Ref. [8].
Two umbrae α and γ are inverse to each other when α + γ ≡ ε. The inverse of the umbraα is denoted by −1.α. Note that α and −1.α are uncorrelated. In dealing with a saturatedumbral calculus, the inverse of an umbra is not unique, but any two umbrae inverse to anygiven umbra are similar.
The symbol α. n denotes the dot-power of α, an auxiliary umbra [8] similar to the productα′α′′ · · · α′′′, where {α′, α′′, . . . , α′′′} is a set of n distinct umbrae, each one similar to theumbra α. We assume α. 0 ≡ u. Properties of the dot-power have been investigated with fulldetails in Ref. [8]. In particular, the moments of the umbra α. n are the n-th power of themoments of the umbra α, that is E[(α. n)k] = an
k for any nonnegative integers k and n.Moments of n.α can be expressed using the notions of integer partition and dot-power
[10]. An expression in terms of exponential Bell polynomials will be given in (2).
The generating function of an umbra The formal power series u + ∑n≥1 αn tn
n! is thegenerating function (g.f.) of the umbra α, and it is denoted by eαt . The notion of umbralequivalence and similarity can be extended coefficientwise to formal power series, that isα ≡ β ⇔ eαt � eβt (see [27] for a formal construction). Note that any exponential formalpower series f (t) = 1+∑
n≥1 antn/n! can be umbrally represented by a formal power series.In fact, if the sequence 1, a1, a2, . . . is umbrally represented by α, then f (t) = E
[eαt
], that
is f (t) � eαt , assuming that we extend E by linearity. In this case, instead of f (t), we writef (α, t) and we say that f (α, t) is umbrally represented by α. Henceforth, when no confusionoccurs, we just say that f (α, t) is the g.f. of α. For example, the g.f. of the augmentationumbra ε is f (ε, t) = 1, while the g.f. of the unity umbra u is f (u, t) = et . The g.f. of thesingleton umbra χ is f (χ, t) = 1 + t, the g.f. of the Bell umbra is f (β, t) = exp
(et − 1
),
and the g.f. of the Bernoulli umbra is f (ι, t) = t/(et − 1
).
The advantage of an umbral notation for g.f.’s is the representation of operations amongg.f.’s through symbolic operations among umbrae. For example, the product of exponentialg.f.’s is umbrally represented by a sum of the corresponding umbrae:
f (α, t) f (γ, t) � e(α+γ )t with f (α, t) � eαt and f (γ, t) � eγ t . (1)
If α is an umbra with g.f. f (α, t), then f (−1.α, t) = 1/ f (α, t). Via (1), the g.f. of n.α isf (n.α, t) = f (α, t)n . Via g.f., the moments of n.α can be expressed as [8]
where Bi, j are the (partial) Bell exponential polynomials and ai are the moments of theumbra α.
The introduction of the g.f.device leads to the definition of new auxiliary umbrae, improv-ing the computational power of the umbral syntax. For this purpose, we could replace R bya suitable polynomial ring having coefficients in R and any desired number of indetermi-nates. Then, an umbra is said to be scalar if the moments are elements of R while it issaid to be polynomial if the moments are polynomials. In this paper, we deal with R[x, y].
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In particular, we define the dot-product of x and α via g.f., i.e. x.α is the auxiliary umbra hav-ing g.f. f (x.α, t) = f (α, t)x . The polynomial umbra x.α has been introduced in Ref. [28],and its properties have been investigated with full details in Ref. [8].
Example 2.4 Bell polynomial umbra. The umbra x.β is the Bell polynomial umbra [8]. Itsfactorial moments are powers of x, that is (x.β)n � xn, and its moments are the exponentialpolynomials, (x.β)n � ∑n
k=0 S(n, k) xk . The g.f.is f (x.β, t) = exp[x(et − 1)].Umbral polynomials Let {qn(x)} be a polynomial sequence of R[x] such that qn(x) =∑n
k=0 qn, k xk, with qn, n �= 0 for all nonnegative integers n. The sequence {qn(α)} con-sists of umbral polynomials with support α such that E[qn(α)] = ∑n
k=0 qn, k ak for anynonnegative integer n. Now suppose q0(x) = 1 and consider an auxiliary umbra η such thatE[ηn] = E[qn(α)], for any nonnegative integer n. In order to underline that the moments of η
depend on those of α, we add the subscript α to the umbra η so that we shall write ηnα � qn(α)
for any nonnegative integer n. If α ≡ x .u, then ηx .u is a polynomial umbra with momentsqn(x), so we shall simply denote it by ηx . Let us consider some simple consequences of thenotations here introduced.
Proposition 2.1 If ηx is a polynomial umbra and α and γ are scalar or polynomial umbrae,then ηα ≡ ηγ ⇔ α ≡ γ .
Proof For any nonnegative integer n, there exist constants cn, k, k = 0, 1, . . . , n such thatxn = ∑n
k=0 cn,k qk(x). Since ηα ≡ ηγ , then qk(α) � qk(γ ) for all nonnegative integers k andso for all nonnegative integers n we have αn � ∑n
k=0 cn, k qk(α) � ∑nk=0 cn, k qk(γ ) � γ n .
The other direction of the proof is straightforward. Proposition 2.2 If ηx and ζx are polynomial umbrae and α is a scalar or polynomial umbra,then ηα ≡ ζα ⇔ ηx ≡ ζx .
Proof Suppose ηα ≡ ζα . Let {qn(x)} be the moments of ηx and let {zn(x)} be the momentsof ζx . For all nonnegative integers n, there exist constants cn, k, k = 0, 1, . . . , n such thatqn(x) = ∑n
k=0 cn, k zk(x). Because qn(α) � zn(α) for all nonnegative integers n, we havecn, k = δn, k for k = 0, 1, . . . , n by which we have qn(x) = zn(x). The other direction of theproof is straightforward. Special auxiliary umbrae A feature of the classical umbral calculus is the construction ofnew auxiliary umbrae by suitable symbolic substitutions. This symbolic substitution relieson the simple observation that E[(n.α)k] is a polynomial in n, as first remarked by Ray [18].In Ref. [8], the explicit expression (2) of these polynomials has been carried out. So in n.α,we can replace the integer n by an umbra γ . From (2), the new auxiliary umbra γ .α hasmoments
where g( j) are the factorial moments of the umbra γ . The auxiliary umbra γ .α is the dot-product of α and γ with g.f. f (γ .α, t) = f (α, log f (γ, t)). Observe that E[γ .α] = g1 a1 =E[γ ] E[α.] In Ref. [8], we have proved that
α.x ≡ α.(xu) ≡ x(α.u) ≡ xα. (4)
Here, we recall some properties that will be often used in the following. If α, γ, η ∈ A then(α + η).γ ≡ α.γ + η.γ and η.(γ .α) ≡ (η.γ ).α. If η.α ≡ η.γ for some umbra η, then
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α ≡ γ . If c ∈ R, then η.(cα) ≡ c(η.α) for any two distinct umbrae α and η. The proofs andsome discussion on these properties can be found in Ref. [8].
In the following, we recall some useful dot-products of umbrae, cf. [8,9] for further details.
Example 2.5 Exponential umbral polynomials. Suppose we replace x with a generic umbraα in the Bell polynomial umbra x.β, [8]. We get the auxiliary umbra α.β, whose factorialmoments are (α.β)n � αn, for all nonnegative integers. The g.f. is f (α.β, t) = f (α, et −1),and the moments are given by the exponential polynomials {�n(x)} with x replaced by α :
(α.β)n � �n(α) �n∑
i=0
S(n, i)αi n = 0, 1, 2, . . . . (5)
Example 2.6 α-partition umbra. The α-partition umbra is the umbra β.α, where β is theBell umbra. Since the factorial moments of β are all equal to 1, Eq. (3) gives
for i = 1, 2, . . . , where Yi are the complete exponential polynomials. The umbra x.β.α
is the polynomial α-partition umbra. Since the factorial moments of x.β are powers of x,
Eq. (3) gives
E[(x.β.α)i
]=
i∑j=1
x j Bi, j (a1, a2, . . . , ai− j+1), (7)
so the g.f. is f (x.β.α, t) = exp[x( f (α, t) − 1)]. One of the most relevant properties of thepolynomial α-partition umbra is (x + y).β.α ≡ x.β.α + y.β.α.
The α-partition umbra β.α plays a crucial role in the umbral representation of the com-position of exponential g.f.’s. Indeed, the composition umbra of α and γ is γ .β.α, withg.f. f (γ .β.α, t) = f [γ, f (α, t) − 1]. From Eq. (7), the moments are E[(γ .β.α)i ] =∑i
j=1 g j Bi, j (a1, a2, . . . , ai− j+1), where g j and ai are moments of the umbra γ and α,respectively. The umbra α<−1> is the compositional inverse of α, with g.f. f (α<−1>, t) =f <−1>(α, t), where f <−1>(α, t) is the compositional inverse of f (α, t) in the sense thatf [α<−1>, f (α, t) − 1] = f [α, f (α<−1>, t) − 1] = 1 + t . So we have α.β.α<−1> ≡α<−1>.β.α ≡ χ . In particular, for the unity umbra, we have
β.u<−1> ≡ u<−1>.β ≡ χ, and β.χ ≡ u ≡ χ .β. (8)
We have also f (u<−1>, t) = 1 + log(1 + t), see [9] for more details.
Example 2.7 α-cumulant umbra. The umbra χ .α is the α-cumulant umbra, having g.f.f (χ .α, t) = 1 + log[ f (α, t)]. The α-cumulant umbra gives rise to an umbral represen-tation theorem for formal power series of logarithmic type (cf. Open problems in Ref. [28]).Properties of cumulant umbrae are investigated in details in Ref. [9].
Example 2.8 α-factorial umbra. The umbra α.χ is the α-factorial umbra, having g.f.f (α.χ, t) = f [α, log(1+t)]. In particular, we have (α.χ)n � (α)n = α(α−1) · · · (α−n+1)
for all integers n ≥ 1. Properties of factorial umbrae are investigated in details in Ref. [9].
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3 Sheffer sequences
Up to now, the umbral calculus we have summarized has been set up in the papers [8,20]and [9]. Now we add one more notion that is a key to construct a symbolic theory of Shefferpolynomials. For the rest of the section, assume γ be an umbra with E[γ ] = g1 �= 0 so thatits g.f. f (γ, t) admits compositional inverse.
Definition 3.1 The adjoint umbra of γ is γ ∗ = β.γ <−1>, the γ <−1>-partition umbra.
The adjoint umbra has g.f. f (γ ∗, t) = exp[ f <−1>(γ, t) − 1]. The name parallels theadjoint of an umbral operator [19] since γ .α∗ gives the umbral composition of γ and α<−1>.
Example 3.1 Adjoint of the singleton umbra χ . The compositional inverse of χ is the umbraχ itself. So we have χ∗ ≡ β.χ<−1> ≡ β.χ ≡ u.
Example 3.2 Adjoint of the unity umbra u. By virtue of equivalence (8), the adjoint of theunity umbra u is u∗ ≡ β.u<−1> ≡ χ .
Example 3.3 Adjoint of the Bell umbra β. We have β∗ ≡ u<−1>. Indeed, β.β.β<−1> ≡ χ
and taking the left-hand side dot-product by χ, we have χ .β.β.β<−1> ≡ χ .χ ≡ u<−1>. Thelast equivalence follows observing that f (χ .χ, t) = 1 + log(1 + t) from Example 2.7 andf (u<−1>, t) = 1+ log(1+ t), see [9] for more details. The result follows recalling the secondequivalence in (8) and β.β<−1> ≡ β∗.
Example 3.4 Adjoint of u<−1>. From Definition 3.1, we have (γ <−1>)∗ ≡ β.γ, so that(u<−1>)∗ ≡ β.u ≡ β.
Note that χ .γ ∗ ≡ γ <−1> and χ .(γ <−1>)∗ ≡ γ. From Definition 3.1, we have
γ .γ ∗ ≡ χ ⇒ β.γ .γ ∗ ≡ β.χ ≡ u. (9)
Proposition 3.1 If γ is an umbra with compositional inverse, then (γ <−1>)∗.γ ∗ ≡ u andγ ∗.(γ <−1>)∗ ≡ u.
Proof Because (γ <−1>)∗ ≡ β.γ, the first equivalence follows because (γ <−1>)∗.β.γ <−1> ≡β.γ .β.γ <−1> ≡ β.χ ≡ u. The second equivalence follows from the first replacing γ <−1>
with γ.
Proposition 3.2 For α, γ ∈ A provided with compositional inverses, we have (α.β.γ )∗ ≡γ ∗.α∗.
Proof By Definition 3.1, we have (α.β.γ )∗ ≡ β.(α.β.γ )<−1>. Since(γ <−1>.β.α<−1>
).β.α.β.γ ≡ χ
and also (α.β.γ )<−1>.β.α.β.γ ≡ χ, we have (α.β.γ )<−1> ≡ γ <−1>.β.α<−1>, so that(α.β.γ )∗ ≡ (β.γ <−1>).(β. α<−1>) ≡ γ ∗.α∗.
Definition 3.2 A polynomial umbra σx is said to be a Sheffer umbra for (α, γ ) if
σx ≡ α + x.γ ∗. (10)
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We denote a Sheffer umbra by σ(α,γ )x in order to make explicit the dependence on α and
γ. The g.f. of σ(α,γ )x is
f[σ
(α,γ )x , t
]= f (α, t) ex [ f <−1>(γ,t)−1]. (11)
Given the umbra γ, any Sheffer umbra is uniquely determined by its moments evaluated at0, since via equivalence (10) we have σ
(α,γ )0 ≡ α.
Example 3.5 Poisson–Charlier polynomials. The polynomials {cn(x; a)} such thatcn(x; a) � {(x.χ − a)/a}n are the Poisson–Charlier polynomials. Indeed, since (x.χ)k �(x)k it is straightforward to prove cn(x; a) = a−n ∑n
k=0
( nk
)(−a)n−k(x)k . Denoting by
ωx,a the polynomial umbra whose moments are cn(x; a), we have ωx,a ≡ (x.χ − a)/a. Theumbra ωx,a is called the Poisson–Charlier polynomial umbra. We show that ωx,a is a Shefferumbra for (−u, χ .a.β). Indeed, we have ωx,a ≡ (x.χ − au)/a ≡ −u + x.(χ/a). Observethat (χ/a) ≡ (χ .a.β)∗. Indeed, (χ .a.β)∗ ≡ (χ .a.β.u)∗ ≡ u∗.(aχ)∗ ≡ χ .(aχ)∗. Sinceχ∗ ≡ u then (aχ)∗ ≡ u/a and (χ .a.β)∗ ≡ χ/a. Thus, ωx,a ≡ −u + x.(χ .a.β)∗.
Theorem 3.1 (Generalized Sheffer identity) A polynomial umbra σx is a Sheffer umbra ifand only if there exists an umbra γ, provided with a compositional inverse, such that
ση+ζ ≡ ση + ζ .γ ∗, for any η, ζ ∈ A. (12)
Proof Let σx be a Sheffer umbra for (α, γ ). From equivalence (10), we have σ(α,γ )η+ζ ≡
equivalence (12) choose the umbra x.u as umbra ζ and the augmentation umbra as umbraη. Observe that σε is a scalar umbra. Thus, equivalence (12) becomes σx ≡ σε + x.γ ∗. ByDefinition 3.2, the polynomial umbra σx is a Sheffer umbra for (σε, γ ).
Corollary 3.1 (The Sheffer identity) A polynomial umbra σx is a Sheffer umbra if and onlyif there exists an umbra γ, provided with a compositional inverse, such that
σx+y ≡ σx + y.γ ∗. (13)
Proof If σx is a Sheffer umbra for (α, γ ), equivalence (13) follows from equivalence (12)choosing the umbra x.u as umbra η and the umbra y.u as umbra ζ. Vice versa, if thepolynomial umbra σx satisfies equivalence (13), it satisfies also equivalence (12), due toProposition 2.2.
Equivalence (13) gives the well-known Sheffer identity. Indeed, setting sn(x + y) =E[σ n
x+y], sk(x) = E[σ kx ] and pn−k(y) = E[(y.γ ∗)n−k] and using the binomial expansion,
we have sn(x + y) = ∑nk=0
( nk
)sk(x) pn−k(y). In the next section, we will prove that the
moments of umbrae such y.γ ∗ are binomial sequences.
Theorem 3.2 (The Expansion Theorem) If σ(α,γ )x is a Sheffer umbra for (α, γ ), then η ≡
(−1.α + σ(α,γ )η ).β.γ for any umbra η.
Proof Replacing x with η in equivalence (10), we obtain σ(α,γ )η ≡ α + η.γ ∗. Take the right
dot-product with β.γ of both sides, then σ(α,γ )η .β.γ ≡ α.β.γ +η. The result follows adding
−1.α.β.γ to both sides of the previous equivalence.
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Theorem 3.3 A polynomial umbra σx is a Sheffer umbra if and only if there exists an umbraγ, provided with compositional inverse, such that
σγ+x.u ≡ χ + σx . (14)
Proof If σx is a Sheffer umbra for (α, γ ), equivalence (14) follows from equivalence (12)choosing x.u as umbra η and the umbra γ as umbra ζ, because γ .γ ∗ ≡ χ. Vice versa, let σx
be a polynomial umbra such that equivalence (14) holds for some umbra γ, provided withcompositional inverse. Set x = 0 in equivalence (14). We have σγ ≡ χ + σ0 ⇒ σγ .β.γ ≡χ .β.γ +σ0.β.γ ≡ γ +σ0.β.γ. Due to Theorem 3.2, the Sheffer umbra for (σ0, γ ) satisfiesσ
(σ0,γ )γ .β.γ ≡ γ + σ0.β.γ, therefore σγ ≡ σ
(σ0,γ )γ and σx ≡ σ
(σ0,γ )x by Proposition 2.2.
Corollary 3.2 A polynomial umbra σx is a Sheffer umbra if and only if there exists an umbraγ, provided with compositional inverse, such that σ k
γ+x .u � σ kx + k σ k−1
x for k = 1, 2, . . . .
Proof Take the k-th moment of both sides in equivalence (14). Theorem 3.4 (Orthogonality) Let {sn(x)} be the moments of a Sheffer umbra for (α, γ ). Thepolynomial sequence {sn(x)} is the unique polynomial sequence such that sn(−1.α.β.γ +k.γ ) � (k.χ)n for all nonnegative integers n, k ≥ 0.
Proof The uniqueness follows from Proposition 2.2. The existence follows from equivalence(10), since when x is replaced by −1.α.β.γ + k.γ, we have −1.α.β.γ .γ ∗ + α + k.γ .γ ∗ ≡−1.α.β.χ + α + k.χ ≡ −1.α + α + k.χ ≡ k.χ for all nonnegative k.
In general, if qn(x) = ∑nk=0 qn,k xk and pn(x) are sequence of polynomials, the umbral
composition of qn(x) and pn(x) is the sequence [19]
qn(p(x)) =n∑
k=0
qn,k pk(x). (15)
If qn(x) and pn(x) are Sheffer sequences, the following theorem holds.
Theorem 3.5 (Umbral composition and Sheffer umbrae) If {qn(x)} are the moments of aSheffer umbra for (α, γ ) and {pn(x)} are the moments of a Sheffer umbra for (η, ζ ), then thepolynomials {qn(p(x))} given in (15) are moments of the Sheffer umbra for (α+η.γ ∗, γ .β.ζ ).
Proof We have E{(α + x.γ ∗)n} = qn(x) = ∑nk=0 qn,k xk . By symbolic substitution of x
with the umbra η + x.ζ ∗, we have
[α + (η + x.ζ ∗).γ ∗]n � qn(η + x.ζ ∗) =n∑
k=0
qn,k (η + x.ζ ∗)k .
Then we have qn(η + x.ζ ∗) � qn(p(x)) given in (15). Recalling Proposition 3.2, we haveζ ∗.γ ∗ � (γ .β.ζ )∗ and
qn(p(x)) � {(α + η.γ ∗) + x.(γ .β.ζ )∗
}n (16)
by which the result follows. Two Sheffer sequences are said to be inverse of each other if and only if their umbral
composition (15) gives the sequence {xn}.
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498 E. Di Nardo et al.
Corollary 3.3 (Inverse of Sheffer sequences) The sequence of moments corresponding to theSheffer umbra for (−1.α.β.γ, γ <−1>) are inverses of the sequence of moments correspondingto the Sheffer umbra for (α, γ ).
Proof From equivalence (16), we have qn(p(x)) = xn for all n ≥ 1 if and only if ζ ∗.γ ∗ ≡ uand α + η.γ ∗ ≡ ε, up to similarity. From the first equivalence, taking into account Prop-osition 3.1, we have ζ ≡ γ <−1>. From the second equivalence, we have η.γ ∗ ≡ −1.α
and η.γ ∗.(γ <−1>)∗ ≡ η ≡ −1.α.(γ <−1>)∗. By recalling that (γ <−1>)∗ ≡ β.γ , the resultfollows.
4 Two special Sheffer umbrae
Among Sheffer umbrae, a special role is played by the associated umbra and the Appellumbra. The associated umbrae are polynomial umbrae whose moments {pn(x)} satisfies thewell-known binomial identity
pn(x + y) =n∑
k=0
(nk
)pk(x) pn−k(y) (17)
for all n = 0, 1, 2, . . . . Polynomial sequences satisfying property (17) are said to be of bino-mial type. Every sequence of binomial type is a Sheffer sequence but most Sheffer sequencesare not of binomial type. The Appell umbrae are polynomial umbrae whose moments {pn(x)}satisfies the identity
d
dxpn(x) = npn−1(x) n = 1, 2, . . . . (18)
Associated umbrae Let us consider a Sheffer umbra for the umbrae (ε, γ ), where γ hascompositional inverse and ε is the augmentation umbra.
Definition 4.1 The polynomial umbra σx ≡ x.γ ∗ is the associated umbra of γ.
The g.f. of x.γ ∗ is f (x.γ ∗, t) = ex[ f <−1>(γ,t)−1], because in Eq. (11) we have f (α, t) =f (ε, t) = 1.
Theorem 4.1 An umbra σ(α,γ )x is a Sheffer umbra for (α, γ ) if and only if σ
(α,γ )−1.α.β.γ+x.u is
the umbra associated with γ.
Proof If σ(α,γ )x is a Sheffer umbra for (α, γ ), then σ
(α,γ )−1.α.β.γ+x.u ≡ −1.α.β.γ .γ ∗+α+x.γ ∗,
by which σ(α,γ )−1.α.β.γ+x.u ≡ x.γ ∗, because β.γ .γ ∗ ≡ u and −1.α+α ≡ ε. Thus, the result fol-
lows from Definition 4.1. Vice versa, let ηx be a polynomial umbra such that η−1.α.β.γ+x.u ≡x.γ ∗. Replacing x with k.γ, we have η−1.α.β.γ+k.γ ≡ k.γ .γ ∗ ≡ k.χ. The result followsfrom Theorem 3.4.
We will say that a polynomial sequence {pn(x)} is associated with an umbra γ if andonly if pn(x) � (x.γ ∗)n, for n = 0, 1, 2, . . . or
pn(k.γ ) � (k.χ)n n, k = 0, 1, 2, . . . . (19)
Theorem 4.2 (Umbral characterization of associated sequences) The sequence {pn(x)} isassociated with the umbra γ if and only if:
pn(ε) � εn for n = 0, 1, 2, . . . (20)
pn(γ + x.u) � pn(x) + n pn−1(x) for n = 1, 2, . . . . (21)
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A symbolic handling of Sheffer polynomials 499
Proof If the sequence {pn(x)} is associated with γ , then pn(ε) � (ε.γ ∗)n � εn for alln = 0, 1, 2, . . . , since ε.α ≡ ε for any umbra α. Equivalence (21) follows from Corol-lary 3.2 choosing a Sheffer umbra for (ε, γ ). Vice versa, if equivalences (20) and (21) hold,we prove by induction that the sequence {pn(x)} satisfies (19). Indeed, by equivalence (20),we have pn(0.γ ) � pn(ε) � 1 if n = 0 otherwise being 0 for n = 1, 2, . . . , j. Suppose thatequivalence (19) holds for k = m
pn(m.γ ) � (m.χ)n n = 0, 1, 2, . . . . (22)
By equivalence (21), we have pn[(m + 1).γ )] � pn(γ + m.γ ) � pn(m.γ ) + npn−1(m.γ )
for n = 1, 2, . . . . Due to induction hypothesis (22), we have pn[(m + 1).γ ] � (m.χ)n +n(m.χ)n−1 � (χ + m.χ)n � [(m + 1).χ]n for n = 1, 2, . . . . Since the sequence {pn(x)}verifies (19), it is associated with γ. Theorem 4.3 (The binomial identity) The sequence {pn(x)} is associated with the umbra γ
if and only if it satisfies the binomial identity (17) for all n = 0, 1, 2, . . . .
Proof If the sequence {pn(x)} is associated with the umbra γ, then identity (17) followsfrom the property (x + y).γ ∗ ≡ x.γ ∗ + y.γ ∗. Vice versa, suppose the sequence {pn(x)}satisfies identity (17). Let ηx be a polynomial umbra such that E[ηn
x ] = pn(x). By identity(17), we have
ηx+y ≡ ηx + η′y (23)
with ηx and η′y uncorrelated polynomial umbrae and E[(η′
y)n] = pn(y) for all nonnegative
integers n. In particular, if we replace y with ε in equivalence (23), then ηx ≡ ηx + η′ε
and hence η′ε ≡ ε. So the polynomials {pn(x)} are such that pn(ε) � εn, i.e. they satisfy
equivalence (20). By induction on equivalence (23), we have
ηx+···+x︸ ︷︷ ︸k
≡ ηx + · · · + η′x︸ ︷︷ ︸
k
,
where the polynomial umbrae on the right-hand side are uncorrelated and similar to ηx . Ifthe x’s are replaced by uncorrelated umbrae similar to any umbra γ, provided with a com-positional inverse, then ηk.γ ≡ k.ηγ . Since E[γ ] �= 0, we can choose an umbra γ such thatηγ ≡ χ thus ηk.γ ≡ k.χ , and the result follows from this last equivalence and equivalence(19). Remark 4.1 Since any polynomial sequence of binomial type can be represented by a poly-nomial umbra x.γ, a polynomial sequence of integral type [3] can be represented by replacingx with n. So the theory of these sequences can be recovered using the tools introduced inRef. [8] (see Open problems in Ref. [28]).
Example 4.1 The umbra x.u is associated with the umbra χ. Indeed, we have χ∗ ≡β.χ<−1> ≡ u and so x.χ∗ ≡ x.u. Therefore the polynomial sequence {xn} is associatedwith the umbra χ. The g.f. is f (x.u, t) = ex t and the binomial identity becomes the well-known (x + y)n = ∑n
k=0
( nk
)xk yn−k .
Example 4.2 The umbra x.u∗ ≡ x.χ is associated with the umbra u. The associated poly-nomial sequence is given by {(x.χ)n} � {(x)n}, see Example 2.8. The g.f. is f (x.u∗, t) =(1 + t)x , and the binomial identity becomes (x + y)n = ∑n
k=0
( nk
)(x)k (y)n−k .
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500 E. Di Nardo et al.
Example 4.3 The umbra x.(u<−1>)∗ ≡ x.β is associated with the umbra u<−1>. The associ-ated polynomial sequence is given by {(x.β)n} � {�n(x)}, where {�n(x)} are the exponen-tial polynomials (5).
Appell umbrae Let us consider a Sheffer umbra for the umbrae (α, χ).
Definition 4.2 The polynomial umbra σx ≡ α + x.u is the Appell umbra of α.
This definition is an updated version in terms of Sheffer sequences of the one given inRef. [20]. By identity (11), the g.f.of the Appell umbra of α is f (α + x.u, t) = f (α, t) ext .
We will say that a polynomial sequence {pn(x)} is an Appell sequence if and only if pn(x) �(α + x.u)n, for n = 0, 1, 2, . . . .
Theorem 4.4 (The Appell identity) The polynomial umbra σx is an Appell umbra for someumbra α if and only if σx+y ≡ σx + y.u.
Proof The result follows immediately, choosing as umbra γ the singleton umbra χ in theSheffer identity.
Corollary 4.1 A polynomial umbra σx is an Appell umbra for some umbra α if and only ifσ n
χ+x.u � σ nx + nσ n−1
x .
Proof The result follows from Corollary 3.2, choosing as umbra γ the singleton umbra χ
Roughly speaking, Corollary 4.1 says that when in the Appell umbra we replace x byχ + x.u, the umbra χ acts as a derivative operator. Indeed, by the binomial expansion, wehave
(α + χ + x.u)n �∑k≥0
(nk
)αn−k (χ + x.u)k
�∑k≥0
(nk
)αn−k
[(x.u)k + k(x.u)k−1
]
� (α + x.u)n +∑k≥0
(nk
)αn−k Dx
[(x.u)k
]
� (α + x.u)n + Dx[(α + x.u)n]
.
Therefore, we have
Dx[(α + x.u)n] � (α + χ + x.u)n − (α + x.u)n
and this is why Corollary 4.1 umbrally expresses Eq. (18).
Example 4.4 Power polynomials. The umbra x.u is the Appell umbra of the augmentation.So the power polynomials {xn} are Appell polynomials.
Example 4.5 Bernoulli polynomials. The Appell umbra for the Bernoulli umbra is ι + x.u.
From the binomial expansion, its moments are the Bernoulli polynomials E[(ι + x.u)n] =∑k≥0
( nk
) Bn−k xk, where Bn are the Bernoulli numbers.
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A symbolic handling of Sheffer polynomials 501
5 Applications
Abel polynomials and Lagrange inversion formula Abel polynomials play a leading rolein the theory of associated sequences of polynomials. The main result of this section is theproof that any sequence of binomial type can be represented as Abel polynomials, heart ofthe paper [21]. Here, we give a very simple proof by introducing the notion of the derivativeof an umbra. The connection between polynomial sequences of binomial type and Abel poly-nomials allows a one line proof of the Lagrange version formula and a series of results, someof which we are going to use in the next paragraph, and others useful in the construction ofan umbral theory of free cumulants [12].
Definition 5.1 The derivative umbra αD of an umbra α is the umbra whose powers are suchthat (αD)n � ∂ααn � nαn−1 for n = 1, 2, . . . .
We have f (αD, t) = 1 + t f (α, t), because eαDt � u + ∑n≥1 nαn−1tn/n!. Note that
E[αD] = 1. In particular, we have(eαDt − u
).k � tk (eαt).k � tke(k.α)t . (24)
Example 5.1 Singleton umbra. The singleton umbra χ is the derivative umbra of the aug-mentation umbra ε, that is εD ≡ χ.
Example 5.2 Bernoulli umbra. We have u ≡ (−1.ι)D. Indeed, we have f (−1.ι, t) =(et − 1)/t so that f [(−1.ι)D, t] = 1 + t (et − 1)/t = et = f (u, t).
Example 5.3 Bernoulli-factorial umbra. We have u<−1> ≡ (ι.χ)D. Indeed, f (ι.χ, t) =log(1 + t)/(elog(1+t) − 1) = log(1 + t)/t so that f [(ι.χ)D, t] = 1 + t log(1 + t)/t =f (u<−1>, t).
Theorem 5.1 (Abel representation of binomial sequences) If γ is an umbra provided witha compositional inverse, then for all x ∈ R
(x.γ ∗
D
)n � x(x − n.γ )n−1, n = 1, 2, . . . . (25)
In the following, we refer to polynomials x(x − n.γ )n−1 as umbral Abel polynomials.
Proof On the basis of Theorem 4.2, the result follows showing that umbral Abel polyno-mials are associated with the umbra γD, i.e. showing that such polynomials satisfy equiva-lences (20) and (21). Since ε(ε − n.γ )n−1 � εn, equivalences (20) are satisfied. Moreover,it is easy to check by simple calculations that (x.u + γD)n − xn � n(x.u + γ )n−1, forn = 1, 2, . . . and more in general p(x.u + γD) − p(x) � p′(x.u + γ ) for any polynomialp(x) ∈ R[A][x], where p′(x) denotes the derivative with respect to x of p(x). In partic-ular for pn(x) = x(x − n.γ )n−1, we have pn(x.u + γD) − pn(x) � p′
n(x.u + γ ). Sincep′
n(x) � n (x−n.γ )n−2 (x−1.γ ) then p′n(x.u+γ ) � n x (x−(n−1).γ )n−2 � n pn−1(x),
and so equivalences (21) are satisfied. Theorem 5.1 includes the well-known Transfer Formula [19]. The following corollary
gives the Lagrange inversion formula.
Corollary 5.1 For any umbra γ, we have(γ <−1>
D
)n � (−n.γ )n−1, n = 1, 2, . . . . (26)
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502 E. Di Nardo et al.
Proof Since χ .β ≡ u then γ <−1>D ≡ χ .β.γ <−1>
D . From equivalence (25), with x replaced byχ, we have (γ <−1>
D )n � (χ .β.γ <−1>D )n � χ(χ−n.γ )n−1, for n = 1, 2, . . . . Because χk+1 �
0 for k = 1, 2, . . . , n − 1, we have χ(χ − n.γ )n−1 � ∑n−1k=0
(n−1
k
)χk+1(−n.γ )n−1−k �
(−n.γ )n−1 by which equivalence (26) follows. The generalization of Theorem 14 required in Ref. [28] (see Open problems) is the fol-
lowing corollary.
Corollary 5.2 If γ is an umbra provided with a compositional inverse, then for all α ∈ Awe have (α.γ ∗
D )n � α(α − n.γ )n−1, for n = 1, 2, . . . .
One more application of Theorem 5.1 is the proof of the following theorem, giving aproperty of Abel polynomials, known as Abel identity.
Theorem 5.2 (Abel identity) If γ ∈ A, then
(x + y)n �∑k≥0
(nk
)y(y − k.γ )k−1(x + k.γ )n−k . (27)
Proof We have e(y.β.γD)t � ∑k≥0 yk(eγDt −u).k/k!. Replace y by y.γ ∗
D . Since γ ∗D .β.γD ≡ u
we have eyt � ∑k≥0(y.γ ∗
D )k(eγDt − u).k/k! � ∑k≥0(y.γ ∗
D )k(tke(k.γ )t )/k! due to (24), andeyt � 1 + ∑
k≥1 y(y − k.γ )k−1(tke(k.γ )t )/k! from Theorem 5.1. Multiplying both sides byext , we have e(x+y)t � 1 + ∑
k≥1 y(y − k.γ )k−1[tke(x+k.γ )t ]/k!. By expanding both sidesof the last equivalence, the result follows immediately because the coefficients of the samepowers of t are similar.
The umbral polynomials in (27)(
nk
)(x + k.γ )n−k (28)
play a leading role. Indeed, let us underline that using Theorem 5.1, equivalence (27) can berewritten as (x + y)n � ∑
k≥0
( nk
)(y.γ ∗
D )k(x + k.γ )n−k . By replacing y with y.β.γD in thislast equivalence and recalling equivalence (9), we recover the following equivalence
(x + y.β.γD)n �∑k≥0
(nk
)yk(x + k.γ )n−k . (29)
By replacing y with u in (29), we have
(x + β.γD)n �∑k≥0
(nk
)(x + k.γ )n−k . (30)
From equivalence (29), we recover the following corollary that gives the umbral expressionof Bell exponential polynomials in (2).
Corollary 5.3 (Umbral representation of Bell exponential polynomials) For all nonnegativen, we have
(x.β.γD)n �∑k≥0
(nk
)(k.γ )n−k xk . (31)
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A symbolic handling of Sheffer polynomials 503
So we call generalized Bell polynomials the polynomials given in (28). These polynomialsprovide a link between Riordan arrays and connection constants, when x is replaced by anumbra α, see next section.
Corollary 5.4 (Closed formulae for Stirling numbers) If ι is the Bernoulli umbra, then theStirling numbers of first and second kind can be umbrally represented as
s(n, k) �(
nk
)(k.ι.χ)n−k and S(n, k) �
(nk
)(−k.ι)n−k for k = 0, 1, . . . , n.
Proof From Example 5.3, we have (x)n � (x.χ)n � (x.β.u<−1>)n � [x.β.(ι.χ)D]n .
The umbral representation of Stirling numbers of first kind follows from comparing the n-th moment of x.β.(ι.χ)D, computed with equivalence (31), with the well-known equality(x)n = ∑
k≥0 s(n, k) xk . For the Stirling numbers of second kind, choose −1.ι as umbra γ
in equivalence (31). From Example 5.2, we have x.β.(−1.ι)D
≡ x.β. Again, the umbralrepresentation of Stirling numbers of second kind follows from comparing the n-th momentof x.β.(−1.ι)
D, computed with equivalence (31), with the umbral equivalence (x.β)n �∑
k≥0 S(n, k)xk, given in Example 2.4. Connection constants and Riordan arrays Theorem 5.1 and equivalence (31) refer to nor-malized binomial polynomials, i.e. sequences {pn(x)} such that p1(x) is monic, since theyinvolve umbrae having first moment equal to 1.
More in general, if one would recover umbral expressions of Sheffer sequences coef-ficients, an additional step is necessary in order to consider umbrae having first momentdifferent from zero and not equal to one.
Proposition 5.1 (Generalized Lagrange inversion formula) For all umbrae γ provided withcompositional inverse and moments {gn}, we have
γ .n (γ <−1>
)n � (−n.γ̃ )n−1 , (32)
where γ̃ is the umbra with moments E[γ̃ n−1
] = gn/(ng1) for n = 1, 2, . . . .
Proof For any umbra γ provided with compositional inverse (that is g1 �= 0), thereexists 1 an umbra α such that γ /g1 ≡ αD . Such an umbra α has moments αn−1 � γ n/(n gn
1 )
for n = 1, 2, . . . and g.f. f (α, t) = [ f (γ, t/g1) − 1]/t. In particular, we have g1α ≡ γ̃ ,
so multiplying by gn−11 both sides of equivalence (26), written for the umbra α, we have
lows since we have (γ /g1)<−1> ≡ χ .g1.β.γ <−1> from equivalence (4) and (α<−1>
D )n �[(γ /g1)
<−1>]n � g1(γ<−1>)n .
The generalization of equivalence (29) to umbrae γ with first moment g1 �= 1 can bestated using the same arguments. Indeed, since (x.β.γ )n � γ .n(x.β.αD)n, we have
(x + y.β.γ )n �n∑
k=0
(nk
)ykγ .k(x + k.γ̃ )n−k . (33)
By setting x = 0 in (33), we have
(y.β.γ )n �∑k≥0
(nk
)ykγ .k(k.γ̃ )n−k, (34)
1 In this case, in the setting of the umbral calculus R must be a field.
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504 E. Di Nardo et al.
and by comparing (34) to (7), we have Bn,k(g1, g2, . . . , gn−k+1) � ( nk
)γ .k[k.γ̃ ]n−k . Using
equivalence (33), it is straightforward to prove the following proposition.
Proposition 5.2 (Umbral representation of Sheffer polynomials) If {sn(x)} are moments ofa Sheffer umbra for (α, γ ), then for all nonnegative integers n
sn(x) �n∑
k=0
(nk
)xkδ.k(α + k.δ̃)n−k with δ ≡ γ <−1>.
An (exponential) Riordan array is a pair (g(t), f (t)) of (exponential) formal power series,where g(t) is an invertible series and f (0) = 0. The pair defines an infinite lower triangulararray (dn,k)0≤k,n<∞ according to the rule dn,k = n-th coefficient of g(t)[ f (t)]k/k!. Rior-dan arrays are known also as recursive matrices [4]. The reader may consult [26] for moreresults on the theory of Riordan arrays. In recent years, this subject has aroused some interest.We mention some results of recent literature that would benefit of the umbral approach, see[1,5,6,14,30]. If sn(x) = ∑n
k=0 sn,k xk is a polynomial of Sheffer type, then from (11) wehave sn,k = n-th coefficient of f (α, t)[ f <−1>(γ, t)−1]k/k!. From Proposition 5.2, we havesn,k � ( n
k
)δ.k(α + k.δ̃)n−k . Hence, the following corollary is proved.
Corollary 5.5 (Umbral representation of exponential Riordan arrays) The elements of theexponential Riordan array ( f (α, t), f <−1>(γ, t) − 1) are umbrally represented by
dn,k �(
nk
)δ.k(α + k.δ̃)n−k with δ ≡ γ <−1>. (35)
Also the connection constants have an umbral representation similar to the elements of aRiordan array (35). Recall that when sn(x) and rn(x) are Sheffer sequences, the constantscn,k in the expression
rn(x) =n∑
k=0
cn,ksk(x) (36)
are known as connection constants. Via umbral syntax, we provide an easy implementableclosed form formula for {cn,k}.
Theorem 5.3 (Umbral representation of connection constants) Let {sn(x)} be the momentsof a Sheffer umbra for (α, γ ) and {rn(x)} be the moments of a Sheffer umbra for (δ, ζ ). Theconnection constants {cn,k} in (36) are such that
cn,k �(
nk
)ξ .k(ς + k.ξ̃ )n−k, with ς ≡ (−1.α.β.γ + δ.β.ζ ).ζ ∗ and ξ ≡ ζ .γ ∗.
(37)
Proof Suppose ηx be a polynomial umbra such that E[ηnx ] = qn(x) = ∑n
k=0 cn,k xk . From(36), we have
δ + x.ζ ∗ ≡ ηα+x.γ ∗ . (38)
In equivalence (38), replace x with −1.α.β.γ + x.β.γ. Due to equivalence (9), the right-hand side of equivalence (38) becomes ηα+(−1.α.β.γ+x.β.γ ).γ ∗ ≡ η−1.α+α+x.β.γ .γ ∗ ≡ ηx.u .
The left-hand side can be simply rewritten as δ + (−1.α.β.γ + x.β.γ ).ζ ∗ ≡ (−1.α.β.γ +
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A symbolic handling of Sheffer polynomials 505
δ.β.ζ ).ζ ∗ + x.β.γ .ζ ∗. Now we have x.β.γ .ζ ∗ ≡ x.(γ <−1>)∗.ζ ∗ ≡ x.(ζ .β.γ <−1>)∗ due toProposition 3.2. Therefore, equivalence (38) can be simply rewritten as
To get the explicit expression of {cn,k} in (36), it is necessary to expand the n-th moment of ηx
in (39) and to get the coefficient of xk . The result follows taking into account Proposition 5.2.
So Riordan arrays and connection constants have a similar umbral representation viageneralized Bell polynomials (28) with x replaced by an umbra α.
Corollary 5.6 The Riordan array for the pair ( f (α, t), f (γ, t) − 1) is the inverse of theRiordan array for the pair (1/ f [α, f <−1>(γ, t) − 1], f <−1>(γ, t) − 1).
Proof From Corollary 5.5, the elements of the Riordan array for the pair ( f (α, t), f (γ, t)−1)
have the form (35) with δ ≡ γ. They correspond to the coefficients of {sn(x)}, moments ofa Sheffer umbra for (α, γ <−1>). The compositional inverses of {sn(x)} are the polynomials∑n
k=0 cn,k xk such that xn = ∑nk=0 cn,ksk(x). The coefficients {cn,k} are the elements of the
inverses of the Riordan arrays for the pair ( f (α, t), f (γ, t) − 1). Since xn are moments of aSheffer umbra for (ε, χ), then {cn,k} are connection constants (37) with ς ≡ −1.α.β.γ <−1>
and ξ ≡ γ. So from Corollary 5.5, the result follows.
References
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