Pp. 101–130. the Classical Umbral Calculus

Lecture Notes of Seminario Interdisciplinare di Matematica Vol. 8(2009), pp. 101–130.

The classical umbral calculus: She↵er sequences Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato

Abstract1. Following the approach of Rota and Taylor [17], we present an innovative theory of She↵er sequences in which the main properties are encoded by using umbrae. This syntax allows us noteworthy computational simpliﬁcations and conceptual clariﬁca- tions in many results involving She↵er sequences. To give an indication of the e↵ectiveness of the theory, we describe applications to the well-known connection constants problem, to Lagrange inversion formula and to solving some recurrence relations.

1. Introduction As well known, many polynomial sequences like Laguerre polynomials, ﬁrst and second kind Meixner polynomials, Poisson-Charlier polynomials and Stirling polynomials are She↵er sequences. She↵er sequences can be considered the core of umbral calculus: a set of tricks extensively used by mathematicians at the begin- ning of the twentieth century. Umbral calculus was formalized in the language of the linear operators by Gian- Carlo Rota in a series of papers (see [15], [16], and [14]) that have produced a plenty of applications (see [1]). In 1994 Rota and Taylor [17] came back to foundation of umbral calculus with the aim to restore, in light formal setting, the computational power of the original tools, heuristical applied by founders Blissard, Cayley and Sylvester. In this new 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 ↵, called an umbra, that is, to represent the sequence 1,a1,a2,... by means of the sequence 1,↵,↵2,... of powers of ↵ via an operator E,resembling the expectation operator of random variables. The second device is that distinct umbrae may represent the same sequence 1,a1,a2,..., as it happens also in probability theory for independent and identically distributed random variables. It is mainly thanks to these devices that the early umbral calculus has had a rigorous and simple formal look.

1Authors’ address: E. Di Nardo, Università degli Studi della Basilicata, Dipartimento di Matematica e Informatica, Via dell’Ateneo Lucano 10, I-85100 Potenza, Italy; e-mail: [email protected] . H. Niederhausen, Florida Atlantic University, Department of Mathematical Sciences, 777 Glades Road, Boca Raton , Florida 33431-0991, USA; e-mail: [email protected] . D. Senato, Universitàdegli Studi della Basilicata, Dipartimento di Matematica e Informatica, Via dell’Ateneo Lucano 10, I-85100 Potenza, Italy; e-mail: [email protected] . Keywords. Umbral calculus, She↵er sequences, connection constants problem, linear recurrences, Lagrange inversion formula. AMS Subject Classification. 05A40, 05A15, 11B83, 11B37. 102 Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato

At first glance, the classical umbral calculus seems just a notation for dealing with exponential generating functions. Nevertheless, this new syntax has given rise noteworthy computational simplifications and conceptual clarifications in di↵erent contexts. Applications are given by Zeilberger [23], where generating functions are computed for many dicult problems dealing with counting combinatorial objects. Applications to bilinear generating functions for polynomial sequences are given by Gessel [7]. Connections with wavelet theory have been investigated in [19] and [20]. In [4], the development of this symbolic computation has produced the theory of Bell umbrae, by which the functional composition of exponential power series has been interpreted in a e↵ective way. On the basis of this result, the umbral calculus has been interpreted as a calculus of measures on Poisson algebras, generalizing compound Poisson processes [4]. A natural parallel with random variables has been further carried out in [5]. In [6], the theory of k-statistics and polykays has been completely rewritten, carrying out a unifying framework for these estimators, both in the univariate and multivariate cases. Moreover, very fast algorithms for computing these estimators have been carried out. Apart from the preliminary paper of Taylor [22], She↵er sequences have not been described in terms of umbrae. Here we complete the picture, giving many examples and several applications. The paper is structured as follows. Section 2 is provided for readers unaware of the classical umbral calculus. We resume terminology, notation and some basic definitions. In Section 3, we introduce the notion of the adjoint of an umbra. This notion is the key to clarify the nature of the umbral presentation of binomial sequences. In Section 4, by introducing umbral polynomials, we stress a feature of the classical umbral calculus, that is the construction of new umbrae by suitable symbolic substitutions. Section 5 is devoted to She↵er umbrae. We introduce the notion of She↵er umbra from which we derive an umbral characterization of She↵er sequences sn(x) . Theorem 5.3 gives the umbral version of the well-known She↵er identity with{ respect} to the associated sequence. Theorem 5.4 gives the umbral version of a second characterization of She↵er sequences, that is s (x) is said to { n } be a She↵er sequence with respect to a delta operator Q,whenQsn(x)=nsn 1(x) for all n 0. In Section 6, the notion of She↵er umbra is used to introducing two special umbrae: the one associated to an umbra, whose moments are binomial sequences, and the Appell umbra, whose moments are Appell polynomials. We easily state their main properties by umbral methods. In the last section, we discuss some topics to which umbral methods can be fruitfully applied. In particular we deal with the connection constants problem, that gives the coecients in expressing a sequence of polynomials s (x) in terms of a di↵erent sequence of polynomials { n } pn(x) and viceversa. We give a very simple proof of the Lagrange inversion formula{ } by showing that all polynomials of binomial type are represented by Abel polynomials. This last result was proved by Rota, Shen and Taylor in [18], but the authors did not make explicit the relations among the involved umbrae, and thus have not completely pointed out the powerfulness of the result. Moreover, the notion of a She↵er umbra brings to the light the umbral connection between binomial sequences and Abel polynomials. This allows us to give a very handy umbral expression for the Stirling numbers of first and second type. Finally, we would like to stress how the connection between She↵er umbrae and Lagrange inversion formula has smoothed the way to an umbral theory of free cumulants [3]. The classical umbral calculus: She↵er sequences 103

In closing, we provide some examples of exact solutions of linear recursions, which beneﬁt of an umbral approach.

2. The classical umbral calculus In the following, we recall terminology, notation and some basic definitions of the classical umbral calculus, as it has been introduced by Rota and Taylor in [17] and further developed in [4] and [5] . An umbral calculus consists of the following data: a) asetA = ↵,,... , 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 i) E[1] = 1; ii) E[↵ij k]=E[↵i]E[j] E[k] for any set of distinct umbrae in A and for···i, j, . . . , k nonnegative··· integers (uncorrelation property); n d) an element ✏ A, called augmentation [14], such that E[✏ ]=0,n, for any nonnegative integer2 n,where 1 , if i = j, = i, j N ; i,j 0 , if i = j, 2 ⇢ 6 e) an element u A, called unity umbra [4], such that E[un] = 1, for any nonnegative integer2 n.

Asequencea0 =1,a1,a2,... in R is umbrally represented by an umbra ↵ when i E[↵ ]=ai , for i =0, 1, 2,... .

The elements ai are called moments of the umbra ↵. Example 2.1. Singleton umbra. The singleton umbra is the umbra such that E[1]=1 ,E[n] = 0 for n =2, 3,... .

The factorial moments of an umbra ↵ are the elements 1 ,n=0, a = (n) E[(↵) ] ,n>0 , ⇢ n where (↵) = ↵(↵ 1) (↵ n + 1) is the lower factorial. n ··· Example 2.2. Bell umbra. The Bell umbra is the umbra such that

E[()n] = 1 for n =0, 1, 2,... . n In [4] we prove that E[ ]=Bn,whereBn is the n-th Bell number, i.e. the number of partitions of a ﬁnite nonempty set with n elements, or the n-th coecient in the Taylor 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 umbrae occurring in p.Ifp and q are two2 umbral polynomials then i) p and q are uncorrelated if and only if their supports are disjoint; ii) p and q are umbrally equivalent if and only if E[p]=E[q], in symbols p q. ' 104 Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato

2.1. Similar umbrae and dot-product. The notion of similarity among umbrae comes in handy in order to manipulate sequences such n n (1) aian i , for n =0, 1, 2,... i i=0 X ✓ ◆ as moments of umbrae. The sequence (1) cannot be represented by using only the umbra ↵ with moments a0 =1,a1,a2,... .Indeed,↵ being correlated to itself, i n i the product aian i cannot be written as E[↵ ↵ ]. So, as it happens for random variables, we need two distinct umbrae having the same sequence of moments. Therefore, if we choose an umbra ↵0 uncorrelated with ↵ but with the same sequence of moments, we have

n n n n i n i n (2) aian i = E ↵ (↵0) = E[(↵ + ↵0) ] . i i i=0 "i=0 # X ✓ ◆ X ✓ ◆ Then the sequence (1) represents the moments of the umbra (↵ + ↵0). In [17], Rota and Taylor formalize this matter by defining an equivalence relation among umbrae. Two umbrae ↵ and are similar when ↵n is umbrally equivalent to n, for all n =0, 1, 2,... in symbols ↵ ↵n n n =0, 1, 2,... . ⌘ , ' Example 2.3. Bernoulli umbra. The Bernoulli umbra ◆ (cf. [17]) satisfies the umbral equivalence ◆ + u ◆.Its ⌘ moments are the Bernoulli numbers Bn, such that n B = B . k k n k 0 ✓ ◆ X Thanks to the notion of similar umbrae, it is possible to extend the alphabet A with the so-called auxiliary umbrae, resulting from operations among similar umbrae. This leads to construct a saturated umbral calculus, in which auxiliary umbrae are handled as elements of the alphabet. It can be shown that saturated umbral calculi exist and that every umbral calculus can be embedded in a saturated umbral calculus [17]. We shall denote by the symbol n.↵ the dot-product of n and ↵, an auxiliary umbra (cf. [17]) similar to the sum ↵0 + ↵00 + + ↵000 where ··· ↵0,↵00,...,↵000 is a set of n distinct umbrae, each one similar to the umbra ↵.So the{ sequence in} (2) is umbrally represented by the umbra 2.↵. We assume that 0.↵ is an auxiliary umbra similar to the augmentation ✏. The next statements follow from the definition of the dot-product. Proposition 2.1. i) n.↵ n. for some integer n =0if and only if ↵ ; ii) if c ⌘R, then n.(c↵) c(n.↵)6 for any nonnegative⌘ integer n; iii) n.(m2.↵) (nm).↵ ⌘m.(n.↵) for any two nonnegative integers n, m; ⌘ ⌘ iv) (n+m).↵ n.↵ +m.↵0 for any two nonnegative integers n, m and any two ⌘ distinct umbrae ↵ ↵0; v) (n.↵ + n.) n.(↵⌘+ ) for any nonnegative integer n and any two distinct umbrae ↵ and⌘ . The classical umbral calculus: She↵er sequences 105

Two umbrae ↵ and are said to be inverse to each other when ↵ + ".We denote the inverse of the umbra ↵ by 1.↵. Note that they are uncorrelated.⌘ Recall that, in dealing with a saturated umbral calculus, the inverse of an umbra is not unique, but any two umbrae inverse to any given umbra are similar. We shall denote by the symbol ↵. n the dot-power of ↵, an auxiliary umbra similar to the product ↵0↵00 ↵000,where ↵0,↵00,...,↵000 is a set of n distinct umbrae, similar to the umbra ↵···. We assume that{ ↵. 0 is an umbra} similar to the unity umbra u. The next statements follow from the deﬁnition of the dot-power. Proposition 2.2. i) If c R, then (c↵). n c n↵. n for any nonnegative integer n =0; ii) (↵. n2). m ↵. nm (↵⌘. m). n for any two nonnegative integers 6 n, m; . (n+m) ⌘ . n ⌘ . m iii) ↵ ↵ (↵0) for any two nonnegative integers n, m and any two ⌘ distinct umbrae ↵ ↵0; iv) (↵. n)k (↵k). n for⌘ any two nonnegative integers n, k. ⌘ By the statement iv), the moments of ↵. n are:

. n k k . n n (3) E[(↵ ) ]=E[(↵ ) ]=ak ,k=0, 1, 2,... for any nonnegative integer n. Hence the moments of the umbra ↵. n are the n-th power of the moments of the umbra ↵. Moments of n.↵ can be expressed using the notions of integer partitions and dot- powers. Recall that a partition of an integer i is a sequence =(1,2,...,t), t where j are weakly decreasing positive integers such that j=1 j = i.The integers j are named parts of .Thelenght of is the number of its parts and r1 r2 P will be indicated by ⌫. A di↵erent notation is =(1 , 2 ,...), where rj is the number of parts of equal to j and r1 +r2 + = ⌫. We use the classical notation i to denote “ is a partition of i”. By using··· an umbral version of the well-known multinomial` expansion theorem, we have

(4) (n.↵)i (n) d ↵ , ' ⌫ i X` r1 r2 where the sum is over all partitions =(1 , 2 ,...) of the integer i,(n)⌫ = 0 for ⌫ >n,

i! 1 . r1 . r2 (5) d = and ↵ =(↵1) (↵2) , r !r ! (1!)r1 (2!)r2 ··· 1 2 ··· ··· with ↵1,↵2,... uncorrelated and similar umbrae.

2.2. The generating function of an umbra. The formal power series tn (6) u + ↵n n! n 1 X is the generating function (g.f.) of the umbra ↵, and it is denoted by e↵t. The notion of umbral equivalence and similarity can be extended coecientwise to formal power series of R[A][[t]] ↵ e↵t et , ⌘ , ' 106 Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato

(see [21] for a formal construction). Note that any exponential formal power series tn (7) f(t)=1+ a n n! n 1 X can be umbrally represented by a formal power series (6) in R[A][[t]]. In fact, if the sequence 1,a1,a2,... is umbrally represented by ↵ then f(t)=E[e↵t]i.e.f(t) e↵t , ' assuming that we extend E by linearity. We denote the formal power series in (7) by f(↵, t) and we will say that f(↵, t) is umbrally represented by ↵. Henceforth, when no confusion occurs, we will just say that f(↵, t) is the g.f. of ↵. For example the g.f. of the augmentation umbra ✏ is f(✏, t) = 1, while the g.f. of the unity umbra u is f(u, t)=et. The g.f. of the singleton umbra is f(, t)=1+t,the g.f. of the Bell umbra is f(,t)=exp(et 1) and the g.f. of the Bernoulli umbra is f(◆, t)=t/(et 1). The advantage of an umbral notation for g.f.’s is the representation of operations among g.f.’s through symbolic operations among umbrae. For example, the product of exponential g.f.’s is umbrally represented by a sum of the corresponding umbrae: (8) f(↵, t) f(,t) e(↵+)t with f(↵, t) e↵t and f(,t) et . ' ' ' Via (8), the g.f. of n.↵ is f(↵, t)n. Note that . n (9) e(n. ↵) t f(t)n e↵t ' ' Via g.f., we have [4] i i (10) E[(n.↵) ]= (n)j Bi,j(a1,a2,...,ai j+1) i =1, 2,... j=1 X where Bi,j are the (partial) Bell exponential polynomials (cf. [12]) and ai are the moments of the umbra ↵. If ↵ is an umbra with g.f. f(↵, t), then

( 1. ↵) t 1 e . ' f(↵, t) The sum of exponential g.f.’s is umbrally represented by a disjoint sum of umbrae. The disjoint sum (respectively disjoint di↵erence) of ↵ and is the umbra ⌘ (respectively ) with moments u, n=0 u, n=0 ⌘n respectively n , ' ( ↵n + n ,n>0 ' ( ↵n n ,n>0 ! in symbols ⌘ ↵+˙ (respectively ↵ ˙ ). By the definition, we have ⌘ ⌘ (↵ ˙ )t f(↵, t) [f(,t) 1] e ± . ± ' 2.3. Polynomial umbrae. The introduction of the g.f. device leads to the definition of new auxiliary umbrae, improving the computational power of the umbral syntax. For this purpose, we could replace R by a suitable polynomial ring having coecients in R and any desired number of indeterminates. Then, an umbra is said to be scalar if the moments are elements of R while it is said to be polynomial if the moments are polynomials. In this paper, we deal with R[x, y]. In particular, The classical umbral calculus: She↵er sequences 107 we define the dot-product of x and ↵ via g.f., i.e. x.↵ is the auxiliary umbra having g.f. e(x.↵) f(↵, t)x . ' Proposition 2.1 still holds, replacing n with x and m with y. Example 2.4. Bell polynomial umbra. The umbra x. is the Bell polynomial umbra. Its factorial moments are powers of x and its moments are the exponential polynomials (cf. [4]) n (x.) xn and (x.)n S(n, k)xk . n ' ' kX=0 Its g.f. is f(x.,t)=exp[x(et 1)]. 2.4. Special auxiliary umbrae. A feature of the classical umbral calculus is the construction of new auxiliary umbrae by suitable symbolic substitutions. In n.↵ replace the integer n by an umbra . From (10), the new auxiliary umbra .↵ has moments i i (11) E[(.↵) ]= g(j)Bi,j(a1,a2,...,ai j+1) i =1, 2,... j=1 X where g(j) are the factorial moments of the umbra . The auxiliary umbra .↵ is called dot-product of ↵ and . The g.f. f(.↵, t) is such that e(. ↵)t [f(t)] e log f(t) g [log f(t)] . ' ' ' Observe that E[.↵]=g1 a1 = E[] E[↵.] The following statements hold. Proposition 2.3. i) ⌘.↵ ⌘. for some umbra ⌘ if and only if ↵ ; ii) if c ⌘R, then ⌘.(c↵) c(⌘.↵) for any two distinct⌘ umbrae ↵ and ⌘; 2 ⌘ iii) if 0, then (↵ + ⌘). ↵. + ⌘.0; iv) ⌘.(.⌘↵) (⌘.).↵. ⌘ ⌘ For the proofs, see [4]. Observe that from property ii) it follows (12) ↵.x ↵.(xu) x(↵.u) x↵. ⌘ ⌘ ⌘ In the following, we recall some useful dot-products of umbrae, whose properties have been investigated with full details in [4] and [5]. Example 2.5. Exponential umbral polynomials. Suppose we replace x with a generic umbra ↵ in the Bell polynomial umbra x.. We get the auxiliary umbra ↵., whose factorial moments are (13) (↵.) ↵n n =0, 1, 2,... . n ' The moments are given by the exponential umbral polynomials (cf. [4]) n (14) (↵.)n (↵) S(n, i)↵i n =0, 1, 2,... . ' n ' i=0 X The g.f. is f(↵.,t)=f(et 1). 108 Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato

Example 2.6. ↵-partition umbra. The ↵-partition umbra is the umbra .↵,where is the Bell umbra (see Example 2.2). Since the factorial moments of are all equal to 1, equation (11) gives i i (15) E[(.↵) ]= Bi,j(a1,a2,...,ai j+1)=Yi(a1,a2,...,ai) j=1 X for i =1, 2,...,whereYi are the complete exponential polynomials [12]. The umbra x..↵ is the polynomial ↵-partition umbra. Since the factorial moments of x. are powers of x, equation (11) gives i i j (16) E[(x..↵) ]= x Bi,j(a1,a2,...,ai j+1) , j=1 X so the g.f. is f(x..↵, t)=exp[x(f(t) 1)]. The ↵-partition umbra .↵ plays a crucial role in the umbral representation of the composition of exponential g.f.’s. Indeed, the composition umbra of ↵ and is the umbra ..↵. The g.f. is f(..↵, t)=f[,f(↵, t) 1]. The moments are i i (17) E[(..↵) ]= gj Bi,j(a1,a2,...,ai j+1) , j=1 X where gj and ai are moments of the umbra and ↵ respectively. From equivalences (4) and (13), we also have

(18) (..↵)i (.) d ↵ ⌫ d ↵ , ' ⌫ ' i i X` X` where d and ↵ are given in (5). < 1> We denote by ↵ the compositional inverse of ↵, i.e. the umbra having g.f. < 1> < 1> < 1> f(↵ ,t) such that f[↵ ,f(↵, t) 1] = f[↵, f(↵ ,t) 1] = 1 + t,i.e. < 1> < 1> f(↵ ,t)=f (↵, t). So we have < 1> < 1> ↵..↵ ↵ ..↵ . ⌘ ⌘ In particular for the unity umbra, we have < 1> < 1> (19) .u u . , ⌘ ⌘ by which the next fundamental equivalences follow (20) . u .. ⌘ ⌘ Since .. .u , recalling i) of Proposition 2.3, the compositional inverse of the singleton⌘ umbra⌘ is the umbra itself. Example 2.7. ↵-cumulant umbra. The umbra .↵ is the ↵-cumulant umbra, having g.f. f(.↵, t) = 1+log[f(t)]. Then, the umbra is the cumulant umbra of u,theumbrau is the cumulant umbra of < 1> ,theumbrau is the cumulant umbra of . Properties of cumulant umbrae are investigated in details in [5]. A special role is held by the cumulant umbra of a polynomial Bell umbra. Indeed, as it has been proved in [5], the (x.)-cumulant umbra has moments all equal to x: (21) (.x.)n x. ' The classical umbral calculus: She↵er sequences 109

Example 2.8. ↵-factorial umbra. n The umbra ↵. is the ↵-factorial umbra, since (↵.) (↵)n for all nonnegative n. The g.f. is f(↵., t)=f[log(1 + t)]. By using the' ↵-factorial umbra, from equivalence (4) we also have (.↵)i () d ↵ (.)⌫ d ↵ , ' ⌫ ' i i X` X` where d and ↵ are given in (5).

3. Adjoint umbrae

Let be an umbra with E[]=g1 = 0 so that the g.f. f(,t) admits compositional inverse. In this section, we study6 some properties of a special partition < 1> umbra, i.e the -partition umbra. As it will be clariﬁed in the following, this is a key umbra in the theory of binomial polynomials. < 1> Deﬁnition 3.1. The adjoint umbra of is the -partition umbra: < 1> ⇤ = . .

The name parallels the adjoint of an umbral operator [14] since .↵⇤ gives the < 1> umbral composition of and ↵ . Example 3.1. Adjoint of the singleton umbra . The inverse of is the umbra itself. So we have < 1> (22) ⇤ . . u. ⌘ ⌘ ⌘ Example 3.2. Adjoint of the unity umbra u. By virtue of equivalence (19), the adjoint of the unity umbra u is < 1> (23) u⇤ .u . ⌘ ⌘ Example 3.3. Adjoint of the Bell umbra . < 1> < 1> We have ⇤ u .Indeed.. and, taking the left-hand side dot-product by⌘ ,wehave ⌘ < 1> < 1> ... . u . ⌘ ⌘ < 1> The result follows recalling equivalence (20) and . = ⇤.

The adjoint umbra has g.f. < 1> f(⇤,t)=exp[f (,t) 1] . In particular the adjoint of the compositional inverse of an umbra is similar to its partition umbra, i.e. < 1> ( )⇤ .. ⌘ From the previous equivalence, we have < 1> < 1> < 1> (24) ( )⇤.. ... . u ⌘ ⌘ ⌘ < 1> and, replacing with ,wehave

(25) ⇤.. u. ⌘ < 1> Example 3.4. Adjoint of u . < 1> The adjoint of the compositional inverse of the unity umbra is (u )⇤ .u . ⌘ ⌘ 110 Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato

Equivalences (24) and (25) may be rewritten in a more useful way. Indeed, we have < 1> < 1> (26) ( )⇤.⇤ ⇤.( )⇤ u. ⌘ ⌘ Note that the dot-product of an umbra ↵ with the adjoint of is the composition < 1> < 1> umbra of ↵ and ,i.e.↵.⇤ ↵.. . In particular we have ⌘ (27) .⇤ ..⇤ . u ⌘ ) ⌘ ⌘ and also < 1> < 1> (28) .⇤ and .( )⇤ . ⌘ ⌘ Proposition 3.1. The adjoint of the composition umbra of ↵ and is the dot- product of the adjoints of and ↵,thatis

(29) (↵..)⇤ ⇤. ↵⇤ . ⌘ Proof. < 1> < 1> < 1> (↵..)⇤ .(↵..) (. ).(. ↵ ) ⇤. ↵⇤ . ⌘ ⌘ ⌘ ⇤ 4. Umbral polynomials

Let qn(x) be a polynomial sequence of R[x] such that qn(x) has degree n for any n :{ } n n 1 qn(x)=qn, nx + qn, n 1x + + qn, 0 . ··· Moreover, let be ↵ an umbra. The sequence qn(↵) consists of umbral polynomials with support ↵ such that { }

E[qn(↵)] = qn, nan + qn, n 1an 1 + + qn, 0 ··· for any nonnegative integer n. Now suppose q0(x) = 1 and consider an auxiliary umbra ⌘ such that n E[⌘ ]=E[qn(↵)] , for any nonnegative integer n. In order to underline that the moments of ⌘ depend on those of ↵, we add the subscript ↵ to the umbra ⌘ so that we shall write ⌘n q (↵) for n =0, 1, 2,... . ↵ ' n If ↵ x.u,then⌘ is a polynomial umbra with moments q (x), so we shall simply ⌘ x.u n denote it by ⌘x. Let us consider some simple consequences of the notations here introduced.

Proposition 4.1. If ⌘x is a polynomial umbra and ↵ and are umbrae both scalar either polynomial, then ⌘ ⌘ ↵ . ↵ ⌘ , ⌘ Proof. For any nonnegative integer n, there exist constants cn, k,k =0, 1,...,n n n such that x = k=0 cn,k qk(x). Since ⌘↵ ⌘ ,thenqk(↵) qk() for all nonnegative integers k and so for all nonnegative⌘ integers n we have' P n n ↵n c q (↵) c q () n . ' n, k k ' n, k k ' kX=0 kX=0 The other direction of the proof is straightforward. ⇤ The classical umbral calculus: She↵er sequences 111

Proposition 4.2. If ⌘x and ⇣x are polynomial umbrae and ↵ is an umbra either scalar or polynomial, then ⌘ ⇣ ⌘ ⇣ . ↵ ⌘ ↵ , x ⌘ x Proof. Suppose ⌘ ⇣ . Let q (x) be the moments of ⌘ and let z (x) be ↵ ⌘ ↵ { n } x { n } the moments of ⇣x. For all nonnegative integers n, there exist constants cn, k,k = 0, 1,...,n such that q (x)= n c z (x). Being q (↵) z (↵) for all non- n k=0 n, k k n ' n negative integers n,wehavecn, k = n, k for k =0, 1,...,n by which we have P qn(x)=zn(x). The other direction of the proof is straightforward. ⇤ n Proposition 4.3. If ⌘x is a polynomial umbra, ↵i i=1 are uncorrelated scalar n { } umbrae and wi i=1 are some weights in R, then { } n n ˙ (30) ⌘ ˙ +i=1.wi..⌘↵i . +i=1.wi..↵i ⌘ m m k Proof. Suppose E[⌘x ]=qm(x)= k=0 qm, kx . Then for all nonnegative integers m,wehave P m m n q (+˙ n .w ..↵ ) q (+˙ n .w ..↵ )k q w ↵k m i=1 i i ' m,k i=1 i i ' m,k i i ' k=0 k=0 i=1 ! Xn m X X w q ↵k +˙ n .w ..q (↵ ) , ' i m,k i ' i=1 i m i i=1 ! X kX=0 due to equivalence (21). The result follows by observing that m +˙ n .w ..q (↵ ) +˙ n .w ..⌘ . i=1 i m i ' i=1 i ↵i ⇥ ⇤ ⇤ We achieve the proof of the next corollary choosing wi = 1 for i =1, 2,...,n in equivalence (30). n Corollary 4.1. If ⌘x is a polynomial umbra and ↵i i=1 are uncorrelated scalar umbrae, then { } n n ˙ ⌘ ˙ +i=1⌘↵i . +i=1↵i ⌘ 5. Sheffer sequences In this section we give the deﬁnition of She↵er umbra by which we recover fundamental properties of She↵er sequences. In the following let us ↵ and be scalar umbrae, with g = E[] = 0. 1 6 Deﬁnition 5.1. A polynomial umbra x is said to be a She↵er umbra for (↵,)if

(31) ( 1.↵ + x.u).⇤ , x ⌘ where ⇤ is the adjoint umbra of .

(↵,) In the following, we denote a She↵er umbra by x in order to make explicit the dependence on ↵ and . We note that if ↵ has g.f. f(↵, t) and has g.f. f(,t), the g.f of ( 1.↵+x.u).⇤ < 1> < 1> xt is the composition of f( ,t)=f (,t) and f( 1.↵+x.u, t)=e /f(↵, t), i.e.

< 1> (↵,) 1 x [f (,t) 1] (32) f( ,t)= e . x f[↵, f < 1>(,t) 1] 112 Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato

(↵,) Theorem 5.1 (The expansion theorem). If x is a She↵er umbra for (↵,), then (33) ⌘ ↵ + (↵,).. ⌘ ⌘ for any umbra ⌘. Proof. Replacing x with ⌘ in equivalence (31), we obtain (↵,) ( 1.↵ + ⌘).⇤ . ⌘ ⌘ Take the right dot product with . of both sides, then (↵,) . . ( 1.↵ + ⌘).⇤. . ( 1.↵ + ⌘) . ⌘ ⌘ ⌘ The result follows adding ↵ to both sides of the previous equivalence. ⇤ (↵,) Theorem 5.2. Let sn(x) be the moments of a She↵er umbra x .Thepoly- nomial sequence s ({x) is} the unique polynomial sequence such that: { n } (34) s (↵ + k.) (k.)n n, k 0 . n ' 8 Proof. From equivalence (31), when x is replaced by ↵ + k.,wehave

(35) ( 1.↵ + ↵ + k.).⇤ k..⇤ k. k 0 ⌘ ⌘ 8 which gives (34). The uniqueness follows from Proposition 4.2. ⇤ We explicitly note that any She↵er umbra is uniquely determined by its moments evaluated at 0, since via equivalence (31) we have (↵,) 1.↵.⇤ . 0 ⌘ (↵,) In Theorem 5.3 we will prove that the moments of the umbra x satisfy the She↵er identity. Example 5.1. Power polynomials. Choosing as umbra ↵ the umbra ✏ and as umbra the umbra , from equivalence (31) we have (✏,) x.u, x ⌘ n since ⇤ u. So the sequence of polynomials x is a She↵er sequence, being ⌘ (✏,) { } moments of the She↵er umbra x . Example 5.2. Poisson-Charlier polynomials. The Poisson-Charlier polynomials are n 1 n n k c (x; a)= ( a) (x) , n an k k kX=0 ✓ ◆ hence x. a n c (x; a) , n ' a ⇢ recalling that (x.)k (x) . Denoting by ! the polynomial umbra whose mo- ' k x,a ments are cn(x; a), we have x. a ! . x,a ⌘ a The classical umbral calculus: She↵er sequences 113

The umbra !x,a is called the Poisson-Charlier polynomial umbra. We show that !x,a is a She↵er umbra for (a.,.a.). Indeed x. a.. ! [ 1.(a.)+x.u]. . x,a ⌘ a ⌘ a So the Poisson-Charlier polynomial umbra !x,a is a She↵er umbra, being (.a.)⇤ . a ⌘

Theorem 5.3 (The She↵er identity). A polynomial umbra x is a She↵er umbra if and only if there exists an umbra ⌘, provided with a compositional inverse, such that

(36) + y.⌘⇤ . x+y ⌘ x Proof. Let x be a She↵er umbra for (↵,). By Deﬁnition 5.1, we have (↵,) ( 1.↵ + x.u + y.u).⇤ x+y ⌘ ⌘ (↵,) ( 1.↵ + x.u).⇤ + y.⇤ x + y.⇤ . ⌘ ⌘ Viceversa, set x = 0 in equivalence (36). We have

(37) + y.⌘⇤ . y ⌘ 0 On the other hand, known the moments of ⌘, there exists an umbra ↵ such that 1.↵.⌘⇤ . Then the polynomial umbra is a She↵er umbra, being ⌘ 0 y y ⌘ ( 1.↵ + y.u).⌘⇤. ⇤ Equivalence (36) gives the well-known She↵er identity, because by using the binomial expansion we have n n (38) sn(x + y)= sk(x) pn k(y) k kX=0 ✓ ◆ n k n k where sn(x + y)=E[x+y], sk(x)=E[x] and pn k(y)=E[(y.⌘⇤) ]. In the next section, we will prove that the moments of umbrae such y.⌘⇤ are binomial sequences. (↵,) Corollary 5.1. If x is a She↵er umbra for (↵,), then (↵,) (↵,) + ⇣.⇤ , ⌘+⇣ ⌘ ⌘ where ⌘ and ⇣ are umbrae.

Theorem 5.4. A polynomial umbra x is a She↵er umbra if and only if there exists an umbra ⌘, provided with compositional inverse, such that (39) + . ⌘+x.u ⌘ x Proof.Ifx is a She↵er umbra for (↵,)then (↵,) ( 1.↵ + x.u + ).⇤ ( 1.↵ + x.u).⇤ + , +x.u ⌘ ⌘ recalling that .⇤ . Then equivalence (39) follows from equivalence (31) choos- ⌘ ing as umbra ⌘ the umbra . Viceversa, let x be a polynomial umbra such that equivalence (39) holds for some umbra ⌘,withE[⌘]=g1 = 0. Set x =0in equivalence (39). We have 6 + ..⌘ ..⌘ + ..⌘ ⌘ + ..⌘. ⌘ ⌘ 0 ) ⌘ ⌘ 0 ⌘ 0 114 Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato

Known the moments of ⌘, there exists an umbra ↵ such that 1.↵.⌘⇤ so that ⌘ 0 ..⌘ ⌘ 1.↵. ⌘ ⌘ (↵,⌘) Due to equivalence (33), also the She↵er umbra for (↵,⌘) is such that ⌘ ..⌘ (↵,⌘) (↵,⌘) ⌘ ⌘ 1.↵, therefore ⌘ and x by Proposition 4.2. ⌘ ⌘ x ⌘ ⇤

Corollary 5.2. A polynomial umbra x is a She↵er umbra if and only if there exists an umbra ⌘, provided with compositional inverse, such that k k k 1 + k for k =1, 2,... . ⌘+x.u ' x x Proof. Take the k-th moment of both sides in equivalence (39). ⇤ n k Now suppose sn(x)= k=0 sn,k x be the moments of a She↵er umbra for (↵,) and rn(x) be the moments of a She↵er umbra for (⌘,⇣). The umbra P [ 1.↵ +( 1.⌘ + x.u).⇣⇤].⇤ [ 1.(↵..⇣ + ⌘)+x.u].⇣⇤.⇤ ⌘ has moments umbrally equivalent to n

(40) sn,krk(x) , kX=0 i.e. the Roman-Rota umbral composition sn(r(x)). So we have proved the following theorem. Theorem 5.5 (Umbral composition and She↵er umbrae). The polynomials given in (40) are moments of the She↵er umbra for (↵..⇣ + ⌘,..⇣). Two She↵er sequences are said to be inverse of each other if and only if their Roman-Rota umbral composition (40) gives the sequence xn . { } Corollary 5.3 (Inverse of She↵er sequences). The sequence of moments corre- < 1> sponding to the She↵er umbra for ( 1.↵.⇤, ) are inverses of the sequence of moments corresponding to the She↵er umbra for (↵,). Proof.Set < 1> ⇣ and ⌘ 1.↵.⇤ , ⌘ ⌘ in (↵..⇣ + ⌘,..⇣). The corresponding She↵er umbra has moments umbrally equivalent to xn. ⇤

6. Two special Sheffer umbrae In this section, we study two special classes of She↵er umbrae: the associated umbra and the Appell umbra. The associated umbrae are polynomial umbrae whose moments p (x) satisﬁes the well-known binomial identity { n } n n (41) pn(x + y)= pk(x) pn k(y) k kX=0 ✓ ◆ for all n =0, 1, 2,... . Every sequence of binomial type is a She↵er sequence but most She↵er sequences are not of binomial type. The concept of binomial type has applications in combinatorics, probability, statistics, and a variety of other ﬁelds. The classical umbral calculus: She↵er sequences 115

The Appell umbrae are polynomial umbrae whose moments pn(x) satisﬁes the identity { } d (42) pn(x)=npn 1(x) n =1, 2,... . dx Among the most notable Appell sequences, besides the trivial example xn , are the Hermite polynomials, the Bernoulli polynomials, and the Euler polynomials.{ } 6.1. Associated umbrae. Let us consider a She↵er umbra for the umbrae (✏,), where has compositional inverse and ✏ is the augmentation umbra.

Deﬁnition 6.1. A polynomial umbra x is said to be the associated umbra of if

x.⇤ , x ⌘ where ⇤ is the adjoint umbra of .

The g.f. of x.⇤ is < 1> x[f (,t) 1] f(x.⇤,t)=e , because in equation (32) we have f(↵, t)=f(✏, t) = 1. The expansion theorem for associated umbrae (cf. Theorem 5.1) is

(43) ⌘ ⌘.⇤... ⌘ (↵,) (↵,) Theorem 6.1. An umbra x is a She↵er umbra for (↵,) if and only if ↵+x.u is the umbra associated to . (↵,) Proof.Ifx is a She↵er umbra for (↵,)then (↵,) ( 1.↵ + ↵.u + x.u).⇤ , ↵+x.u ⌘ by which (↵,) x.⇤ . ↵+x.u ⌘ (↵,) From Deﬁnition 6.1, the umbra ↵+x.u is the umbra associated to . Viceversa, let ⌘x be a polynomial umbra such that

⌘ x.⇤ . ↵+x.u ⌘ Replacing x with k.,wehave

⌘ k..⇤ k.. ↵+k. ⌘ ⌘ The result follows by equivalence (35). ⇤ We will say that a polynomial sequence pn(x) is associated to an umbra if and only if { } n p (x) (x.⇤) ,n=0, 1, 2,... n ' or (44) p (k.) (k.)n n, k =0, 1, 2,... . n ' Theorem 6.2 (Umbral characterization of associated sequences). The sequence p (x) is associated to the umbra if and only if: { n } (45) p (✏) ✏n for n =0, 1, 2,... n '

(46) pn( + x.u) pn(x)+npn 1(x) for n =1, 2,... . ' 116 Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato

Proof.Ifthesequence p (x) is associated to then { n } n n p (✏) (✏.⇤) ✏ n ' ' for all n =0, 1, 2,... . Equivalence (46) follows from Corollary 5.2 choosing as umbra ↵ the umbra ✏. Viceversa, if equivalences (45) and (46) hold, we prove by induction that the sequence pn(x) satisﬁes (44). Indeed, by equivalence (45) we have { } 0 ,n=1, 2,...,j , pn(0.) pn(✏) ' ' ( 1 ,n=0. Suppose that equivalence (44) holds for k = m (47) p (m.) (m.)n n =0, 1, 2,... . n ' By equivalence (46), we have

pn[(m + 1).)] pn( + m.) pn(m.)+npn 1(m.) n =1, 2,... . ' ' Due to induction hypothesis (47), we have n n 1 n n p [(m+1).] (m.) +n(m.) (+m.) [(m+1).] n =1, 2,... . n ' ' ' Since the sequence p (x) veriﬁes (44), it is associated to . { n } ⇤

Theorem 6.3 (The binomial identity). The sequence pn(x) is associated to the umbra if and only if { } n n (48) pn(x + y)= pk(x) pn k(y) k kX=0 ✓ ◆ for all n =0, 1, 2,... .

Proof.Ifthesequence pn(x) is associated to the umbra , then identity (48) follows from the property{ }

(x + y).⇤ x.⇤ + y.⇤ . ⌘ Viceversa, suppose the sequence pn(x) satisﬁes identity (48). Let ⌘x be a polynomial umbra such that { } n E[⌘x ]=pn(x) . By identity (48), we have

(49) ⌘ ⌘ + ⌘0 x+y ⌘ x y with ⌘x similar to ⌘x0 and uncorrelated. In particular, if we replace y with ✏ in equivalence (49), then ⌘x ⌘x + ⌘✏0 and hence ⌘✏0 ✏. So the polynomials pn(x) n⌘ ⌘ { } are such that pn(✏) ✏ , i.e. they satisfy equivalence (45). By induction on equivalence (49), we have'

⌘x + + x ⌘x + + ⌘x0 ··· ⌘ ··· k k where the polynomial umbrae| on{z the right-hand} | {z side are} uncorrelated and similar to ⌘x.Ifthex’s are replaced by uncorrelated umbrae similar to any umbra , provided of compositional inverse, then ⌘ k.⌘ . k. ⌘ The classical umbral calculus: She↵er sequences 117

Since E[] = 0, we can choose an umbra such that 6 ⌘ ⌘ thus (50) ⌘ k. k. ⌘ and the result follows from equivalences (50) and (44). ⇤ Example 6.1. The umbra x.u is associated to the umbra . Indeed, the polynomial n < 1> sequence x is associated to the adjoint umbra ⇤ . u. The g.f. is { } ⌘ ⌘ xk f(x.u, t)= tk k! k 0 X and the binomial identity becomes n n n k n k (x + y) = x y . k kX=0 ✓ ◆ Example 6.2. The umbra x.u⇤ x. is associated to the umbra u. The associated polynomial sequence is given by⌘(x.)n (x) , see Example 2.8. The g.f. is { }'{ n} x f(x.u⇤,t)=(1+t) and the binomial identity becomes n n (x + y)n = (x)k (y)n k . k kX=0 ✓ ◆ < 1> < 1> Example 6.3. The umbra x.(u )⇤ x. is associated to the umbra u . ⌘ n The associated polynomial sequence is given by (x.) n(x) ,where n(x) are the exponential polynomials (14). { }'{ } { }

Now, suppose p (x) be the polynomial sequence associated to an umbra { n } with g.f. f(,t) and qn(x) be the polynomial sequence associated to an umbra ⇣ with g.f. f(⇣,t), i.e. { } < 1> n < 1> n p (x) (x.. ) and q (x) (x..⇣ ) n ' n ' for n =0, 1, 2,... . On the other hand, due to Proposition 3.1, we have < 1> < 1> < 1> < 1> (x.. )..⇣ x.(. ).(.⇣ ) x.⇤.⇣⇤ x.(⇣..)⇤ . ⌘ ⌘ ⌘ Following Roman-Rota notation, the umbra x.(⇣..)⇤ has moments n

(51) qn(p(x)) = qn,kpk(x) . kX=0 This proves the next theorem. Theorem 6.4 (Umbral composition of associated sequences). The polynomial sequence (51) is associated to the compositional umbra of ⇣ and . Corollary 6.1 (Inverse of associated sequences). The polynomial sequence asso- < 1> ciated to the umbra is inverse of the polynomial sequence associated to the umbra . 118 Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato

< 1> Proof. Choosing as umbra the umbra ⇣ in the previous theorem, we have

x.⇣⇤..⇣ x.u. ⌘ ⇤

Example 6.4. Choose as umbra ⇣ the umbra u.Thenx.u⇤..u x.. x.u and so ⌘ ⌘ n n x = S(n, k)(x)k , kX=0 which is the well-known formula giving powers in terms of lower factorials.

Finally, via Proposition 3.1, it is easy to prove the following recurrence formula

n+1 < 1> n (x.⇤) x [(x + ).⇤] . ' 6.2. Appell umbrae. In this section, we consider a second kind of special She↵er umbra, that is a She↵er umbra for (↵,).

Deﬁnition 6.2. A polynomial umbra x is said to be the Appell umbra of ↵ if 1.↵ + x.u. x ⌘ By equivalence (32), being f(, t)=1+t, the g.f. of ( 1.↵ + x.u)is 1 f( 1.↵ + x.u, t)= ext . f(↵, t)

We will say that a polynomial sequence pn(x) is an Appell sequence if and only if { } p (x) ( 1.↵ + x.u)n ,n=0, 1, 2,... . n ' The expansion theorem for Appell umbrae (cf. Theorem 5.1) easily follows by observing that ⌘ ↵ +( 1.↵ + ⌘) . ⌘ Theorem 6.5 (The Appell identity). The polynomial umbra x is an Appell umbra for some umbra ↵ if and only if + y. x+y ⌘ x Proof. The result follows immediately, choosing as umbra the singleton umbra in the She↵er identity. ⇤

Corollary 6.2. A polynomial umbra x is an Appell umbra for some umbra ↵ if and only if

n n n 1 (52) + n (x) . +x.u ' x Proof. The result follows from Corollary 5.2, choosing as umbra the singleton umbra . ⇤ The classical umbral calculus: She↵er sequences 119

Roughly speaking, Corollary 6.2 says that, when in the Appell umbra we replace x by + x.u,theumbra acts as a derivative operator. Indeed by the binomial expansion, we have

n n n k k ( 1.↵ + + x.u) ( 1.↵) ( + x.u) ' k ' k 0 ✓ ◆ X n n k k k 1 ( 1.↵) [(x.u) + k(x.u) ] ' k ' k 0 ✓ ◆ X n n n k k ( 1.↵ + x.u) + ( 1.↵) D [(x.u) ] . ' k x k 0 ✓ ◆ X So equivalence (52) umbrally expresses equation (42). Theorem 6.6 (The multiplication theorem). For any constant c R, we have 2 1.↵ +(cx).u c( 1.↵ + x.u)+(c 1).↵. ⌘ Proof.Wehave c( 1.↵ + x.u)+(c 1).↵ c.↵ +(cx).u + c.↵ 1.↵ 1.↵ +(cx).u. ⌘ ⌘ ⇤ Example 6.5. Bernoulli polynomials. The Appell umbra for the inverse of the Bernoulli umbra is ◆ + x.u. From the binomial expansion, its moments are the Bernoulli polynomials

n n k E[(◆ + x.u) ]= Bn k x , k k 0 ✓ ◆ X where Bn are the Bernoulli numbers.

7. Topics in umbral calculus In this section, we apply the language of umbrae to some topics which beneﬁt from this approach. In particular we discuss the well-known connection constants problem, the Lagrange inversion formula, and we solve some recurrence relations to give an indication of the e↵ectiveness of the method.

7.1. The connection constants problem. The connection constants problem consists in determining the connection constants cn,k in the expression n sn(x)= cn,krk(x) , kX=0 where sn(x) and rn(x) are sequences of polynomials. When sn(x) and rn(x) are She↵er sequences, umbrae provide an easy solution to this problem. Indeed, suppose ⌘x be a polynomial umbra such that n n k E[⌘x ]=qn(x)= cn,kx . kX=0 The theorem we are going to prove states that ⌘x is a She↵er umbra whenever sn(x) and rn(x) are She↵er sequences. 120 Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato

Theorem 7.1. If ⌘x is a polynomial umbra such that

(53) ( 1.↵ + x.u).⇤ ⌘( 1.+x.u).⇣ , ⌘ ⇤ < 1> then ⌘ is a She↵er umbra for (( 1.↵).⇣⇤,..⇣ ). x Note that equivalence (53) is the way to transform the She↵er umbra ( 1. + x.u).⇣⇤ in the She↵er umbra ( 1.↵ + x.u).⇤ by using the polynomial umbra ⌘ . x Proof. In equivalence (53), replace x with + x..⇣. The right-hand side of equivalence (53) becomes

⌘[ 1.+(+x..⇣).u].⇣ ⌘x..⇣.⇣ ⌘x.u ⇤ ⌘ ⇤ ⌘ due to equivalence (27), whereas the left-hand side gives

(54) [ 1.↵ +( + x..⇣).u].⇤ ( 1.↵).⇤ + x..⇣.⇤ . ⌘ By Proposition 3.1, we have < 1> < 1> x..⇣.⇤ x.(⇣ )⇤.⇤ x.(..⇣ )⇤ ⌘ ⌘ and by equivalence (26) < 1> ( 1.↵).⇤ ( 1.↵).u.⇤ ( 1.↵).⇣⇤.(⇣ )⇤.⇤ . ⌘ ⌘ Replacing ( 1.↵).⇤ and x..⇣.⇤ in equivalence (54), we have < 1> (55) [ 1.↵ +( + x..⇣).u].⇤ [( 1.↵).⇣⇤ + x.u].(..⇣ )⇤ ⌘ and so equivalence (53) returns < 1> (56) ⌘ [( 1.↵).⇣⇤ + x.u].(..⇣ )⇤ . x ⌘ The result follows from Deﬁnition 5.1. ⇤ To get an explicit expression of the connection constants, we can use equivalence (18) to expand the n-th moment of ⌘x in (56) n ⌫ < 1> ⌘ [( 1.↵).⇣⇤ + x.u] d (⇣.. ) . x ' n X` k The connection constants cn,k are the coecient of x in the previous equivalence. Example 7.1. Consider n

cn(x; b)= cn,k ck(x; a) kX=0 where cn(x; a) and cn(x; b) are Poisson-Charlier polynomials. As shown in Example 5.2, cn(x; b) are the moments of a She↵er umbra for (b.,.b.) and cn(x; a) are the moments of a She↵er umbra for (a.,.a.). By equivalence (56), we have < 1> ⌘ [(a. b.).(.a.)⇤ + x.u].[(.b.)..(.a.) ]⇤ x ⌘ or, equivalently, via equivalence (55)

⌘ ( b. + a. + x...a.).(.b.)⇤ . x ⌘ Observe that x...a. xa.,so ⌘ (57) ( b. + a. + x...a.).(.b.)⇤ (a b + xa)..(.b.)⇤ . ⌘ Moreover (.b.)⇤ (.b..u)⇤ u⇤.(b)⇤ .(b)⇤, from which ⌘ ⌘ ⌘ .(.b.)⇤ ..(b)⇤ (b)⇤ . ⌘ ⌘ The classical umbral calculus: She↵er sequences 121

Since ⇤ u then (b)⇤ u/b and ⌘ ⌘ u .(.b.)⇤ . ⌘ b Replacing this last result in equivalence (57), we have a b + xa ⌘ . x ⌘ b The binomial expansion gives n k n a n b c = 1 . n,k k b a ✓ ◆ ⇣ ⌘ ✓ ◆ 7.2. Abel polynomials and Lagrange inversion formula. Abel polynomials play a leading role in the theory of associated sequences of polynomials. The main result of this section is the proof that any sequence of binomial type can be represented as Abel polynomials, heart of the paper [18]. The proof given in [18] was a hybrid, based both on the early Roman-Rota version of the umbral calculus and the last version, introduced by Rota-Taylor. Here, we give a very simple proof by introducing the notion of the derivative of an umbra.

Deﬁnition 7.1. The derivative umbra ↵D of an umbra ↵ is the umbra whose moments are: n n n 1 (↵ ) @ ↵ n↵ for n =1, 2,... . D ' ↵ ' We have

f(↵D ,t)=1+tf(↵, t) , since n ↵ t n 1 t ↵t e D u + n↵ u + te . ' n! ' n 1 X In particular, we have .k .k (58) e↵D t u tk e↵t tke(k.↵)t . ' ' Note that E[↵D ] = 1. Example 7.2. Singleton umbra. The singleton umbra is the derivative umbra of the augumentation umbra ✏, that is ✏ . D ⌘ Example 7.3. Bernoulli umbra. We have u ( 1.◆) .Indeed,wehave ⌘ D et 1 f( 1.◆, t)= t so that t e 1 t f[( 1.◆)D ,t]=1+t = e = f(u, t) . t Example 7.4. Bernoulli-factorial umbra. < 1> We have u (◆.)D .Indeed ⌘ log(1 + t) log(1 + t) f(◆., t)= = elog(1+t) 1 t 122 Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato so that

log(1 + t) < 1> f[(◆.) ,t]=1+t = 1 + log(1 + t)=f(u ,t) . D t Theorem 7.2 (Abel representation of binomial sequences). If is an umbra provided with a compositional inverse, then n n 1 (59) (x.⇤ ) x(x n.) ,n=1, 2,... D ' for all x R. 2 n 1 In the following, we refer to polynomials x(x n.) as umbral Abel polynomials. Proof. On the basis of Theorem 6.2, the result follows showing that umbral Abel polynomials are associated to the umbra , i.e. showing that such polynomials n 1 n satisfy equivalences (45) and (46). Since ✏(✏ n.) ✏ , equivalences (45) are satisﬁed. Moreover, it is easy to check by simple calculations' that n n n 1 (x.u + ) x n(x.u + ) ,n=1, 2,... D ' and more in general d p(x.u + ) p(x) p(x.u + ) D ' dx n 1 for any polynomial p(x) R[A][x]. In particular for pn(x)=x(x n.) ,we have 2 d p (x.u + ) p (x) p (x.u + ) . n D n ' dx n We state equivalences (46) by proving that d pn(x.u + ) npn 1(x) . dx ' To this aim, we have

d n 2 p (x) n(x n.) (x 1.) dx n ' so d n 2 pn(x.u + ) nx(x (n 1).) npn 1(x) . dx ' ' ⇤ According to Corollary 6.1, the inverses of umbral Abel polynomials with respect to the Roman-Rota umbral composition are the moments of x..D . An umbral expression of the inverses of umbral Abel polynomials will be given in Corollary 7.2. < 1> Since the g.f. of x.⇤ is exp[x (f (D ,t) 1)], we have D k < 1> t k 1 exp[x (f ( ,t) 1)] 1+ [x(x k.)] , D ' k! k 1 X which is the g.f. of umbral Abel polynomials. Theorem 7.2 includes the well-known Transfer Formula. In the following we state various results usually derived by Transfer Formula. We start with the Lagrange inversion formula. The classical umbral calculus: She↵er sequences 123

Corollary 7.1. For any umbra , we have < 1> n n 1 (60) ( ) ( n.) ,n=1, 2,... . D ' < 1> < 1> Proof.Since. u then D ..D . From equivalence (59), with x replaced by ,wehave⌘ ⌘ < 1> n < 1> n n 1 ( ) (.. ) ( n.) ,n=1, 2,... . D ' D ' Since k+1 0 for k =1, 2,...,n 1, we have ' n 1 n 1 n 1 k+1 n 1 k n 1 ( n.) (n.) (n.) ' k ' kX=0 ✓ ◆ by which equivalence (60) follows. ⇤ Remark 7.1. Theorem 7.2 is referred to normalized binomial polynomials, i.e. sequences pn(x) such that p1(x) is monic. This is why the Lagrange inversion formula (60){ refers} to umbrae having ﬁrst moment equal to 1. More in general, if one would consider umbrae having ﬁrst moment di↵erent from zero, one step more is necessary. By way of an example, we do this for the Lagrange inversion formula. For any umbra such that E[]=g = 0, there exists2 an umbra ↵ such that 1 6 /g1 ↵D . Indeed such an umbra ↵ has moments ⌘ n n 1 ↵ n n =1, 2,... ' ng1 and g.f. f(↵, t)=[f(, t/g1) 1]/t. In particular g1↵ ,where is the umbra introduced in [4] having moments ⌘ g E[ n]= n+1 n =0, 1,... , g1(n + 1) n n 1 with E[ ]=gn,n=1, 2,... . Multiplying for g1 both sides of equivalence (60), written for the umbra ↵,wehave n 1 < 1> n n 1 n 1 n 1 n 1 g (↵ ) g ( n.↵) [ n.(g ↵)] ( n. ) 1 D ' 1 ' 1 ' < 1> for n =1, 2,... .Being.. , recalling equivalence (12) and . u,we have ⌘ ⌘ < 1> < 1> < 1> g1.. .g1.. ...g1.. . g1 , g1 ⌘ , g1 ⌘ So, we have < 1> < 1> .g .. g ⌘ 1 ✓ 1 ◆ and from equivalence (18) we have < 1> n < 1> n < 1> n (.g .. ) g ( ) . g ' 1 ' 1 "✓ 1 ◆ # Finally, being < 1> n n 1 < 1> n n 1 n < 1> n g (↵ ) g g ( ) 1 D ' 1 g ' 1 "✓ 1 ◆ #

2In this case, in the setting of the umbral calculus R must be a ﬁeld. 124 Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato we have .n < 1> n n 1 (61) ( ) ( n. ) . ' This last equivalence is the generalized Lagrange inversion formula.

The Lagrange inversion formula (60) allows us to express Stirling numbers of first kind in terms of the Bernoulli umbra. An analogous result was proved by Rota and Taylor in [17] via Nörlund sequences. n 1 Proposition 7.1. If ◆ is the Bernoulli umbra and s(n, 1) = ( 1) (n 1)! for n =1, 2,... are the Stirling numbers of first kind, then n 1 (62) s(n, 1) (n.◆) ,n=1, 2,... . ' < 1> < 1> Proof. From Example 7.3, we have u ( 1.◆)D ,where◆ is the Bernoulli umbra. From equivalence (60), we have ⌘ n < 1> n < 1>) n 1 n 1 (u ) ( 1.◆) [ n.( 1.◆)] (n.◆) ,n=1, 2,... . ' D ' ' h i < 1> Equivalence (62) follows recalling that 1 + log(1 + t) is the g.f. of u and that

1 tn 1 + log(1 + t)=1+ s(n, 1) n! n=1 X with s(k, 1) the Stirling numbers of ﬁrst kind. ⇤ One more application of Theorem 7.2 is the proof of the following proposition, giving a property of Abel polynomials, known as Abel identity. Proposition 7.2 (Abel identity). If A, then 2 n n k 1 n k (63) (x + y) y(y k.) (x + k.) . ' k k 0 ✓ ◆ X Proof. Recall that (eD t u).k e(y..D )t yk . ' k! k 0 X Replace y by y.⇤ .Since⇤ ..D u we have D D ⌘ D t .k k (k.)t yt k (e u) k t e (64) e (y.D⇤ ) (y.D⇤ ) ' k! ' k! ' k 0 k 0 X X k (k.)t k 1 t e (65) y(y k.) ' k! k 0 X where the second equivalence in (64) follows from (58), and equivalence (65) follows from Theorem 7.2. Multiplying both sides by ext,wehave k (x+k.)t (x+y)t k 1 t e (66) e y(y k.) . ' k! k 0 X n (n) (x+y)t (n) Since (x + y) = Dt [e ]t=0,whereDt [ ]t=0 is the n-th derivative with respect to t evaluated at t = 0, the result follows· taking the n-th derivative with The classical umbral calculus: She↵er sequences 125 respect to t of the right-hand side of (66) and evaluating it at t = 0. Indeed, by using the binomial property of the derivative operator we have

k (n) k (x+k.)t n (j) k (n j) (x+k.)t D [t e ] D [t ] D [e ] t ' j t t j=0 X ✓ ◆ and, setting t = 0, we have

(n) k (x+k.)t n k D [t e ] (n) (x + k.) t t=0 ' k by which equivalence (63) follows. ⇤ Setting x = 0 in equivalence (63), we obtain

n n n k k 1 (67) y (k.) y(y k.) . ' k k 0 ✓ ◆ X This proves the following polynomial expansion theorem in terms of Abel polynomials. Proposition 7.3. If p(x) R[x], then 2 k 1 y(y k.) p(x) p(k)(k.) . ' k! k 0 X The following corollary gives the umbral expression of the Bell exponential polynomials in (16). Corollary 7.2 (Umbral representation of Bell exponential polynomials). For all nonnegative n, we have

n n n k k (68) (x.. ) (k.) x . D ' k k 0 ✓ ◆ X Proof. From equivalence (67) and by using Theorem 7.2, we have

n n n k k (69) y (k.) (y.⇤ ) . ' k D k 0 ✓ ◆ X

Replace y with x..D .Wehave

n n n k k n n k k (x.. ) (k.) (x.. .⇤ ) (k.) x , D ' k D D ' k k 0 ✓ ◆ k 0 ✓ ◆ X X by which the result follows immediately recalling equivalence (26) for D . ⇤ The generalization of equivalence (68) to umbrae with ﬁrst moment g1 di↵erent from zero can be stated by using the same arguments given in Remark 7.1:

n .n n n .n n k k (x..) (x..↵ ) (k.↵) x ' D ' k ' k 0 ✓ ◆ X n .k n k k n .k n k k [k.(g ↵)] x [k. ] x . ' k 1 ' k k 0 ✓ ◆ k 0 ✓ ◆ X X 126 Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato

Example 7.5. Stirling numbers of second kind.

In equivalence (68), choose 1.◆ as umbra ,thenx..( 1.◆)D x. (see Example 7.3). Comparing equivalence (68) with (x.)n S(n, k)⌘xk,where S(n, k) ' k 0 { } are Stirling numbers of second kind (see [4]), we have P n n k S(n, k) ( k.◆) k =0, 1,...,n. ' k ✓ ◆ This last equivalence was already proved by Rota and Taylor in [17] through a di↵erent approach.

The umbral version of Stirling numbers of first kind is given in the following proposition. Proposition 7.4. If s(n, k) are Stirling numbers of first kind, then { } n n k (70) s(n, k) (k.◆.) k =0, 1,...,n. ' k ✓ ◆ Proof. Recalling Example 7.4, we have n < 1> n n (x) (x.) (x..u ) [x..(◆.) ] . n ' ' ' D The result follows from equivalence (68), being (x.)n s(n, k) xk . ' k 0 X ⇤ 7.3. Solving recursions. In many special combinatorial problems, the hardest part of the solution may be the discovery of an e↵ective recursion. Once a recursion has been established, She↵er polynomials are often a simple and general tool for finding answers in closed form. Main contributions in this respect are due to Niederhausen [8, 9, 10]. Further contributions are given by Razpet [11] and Di Bucchianico and Soto y Koelemeijer [2]. Example 7.6. Suppose we are asked to solve the di↵erence equation

(71) sn(x + 1) = sn(x)+sn 1(x) 1 under the condition 0 sn(x) dx = 1 for all nonnegative integers n. Equation (71) ﬁts the She↵er identity (38) if we set y = 1, choose the sequence p (x) such that R n p (x)=1,p (1) = 1 and p (1) = 0 for all n 2 and consider the{ She↵er} sequence 0 1 n n!sn(x) .Thesequence pn(x) is associated to the umbra , so we are looking { } { } (↵,) n for solutions of (71) such that n!sn(x) (x ) with ⇤ ,i.e. u (cf. 1 ' ⌘ ⌘ Example 3.2). The condition 0 sn(x) dx = 1 can be translated in umbral terms by looking for an umbra such that E[s ()] = 1 for all nonnegative integers n.Such R n an umbra has g.f. 1 et 1 ext dx = t Z0 so that 1.◆,with◆ the Bernoulli umbra. Therefore, the umbra ↵ satisﬁes the following⌘ identity ( 1.↵ + 1.◆). u. ⌘ The classical umbral calculus: She↵er sequences 127

Due to statement i) of Proposition 2.3, being . u,wehave 1.↵+ 1.◆ and so ↵ 1.(◆+). Solutions of (71) are moments of⌘ the She↵er umbra (◆++⌘x.u). normalized⌘ by n!. Example 7.7. Suppose we are asked to solve the di↵erence equation

(72) sn(x)=sn(x 1) + sn 1(x) that satisﬁes the initial condition n 1 (73) s (1 n)= s (n 2 i) for all n 1 ,s( 1) = 1 . n i 0 i=0 X If we rewrite (72) as sn(x 1) = sn(x) sn 1(x), we note that this equation ﬁts the She↵er identity (38) if we set y = 1, choose the sequence pn(x) such that p (x) = 1, p ( 1) = 1 and p ( 1) = 0 for all n 2, and consider{ } the She↵er 0 1 n sequence n!sn(x) . In particular, from (36) we have 1.⇤ so 1. { } 1 ⌘ ⇤⌘ which has g.f. f( 1. , t)=(1 t) . Suppose to set u = 1. .Wehave E[un]=n! for alln 1 and x.u x. with g.f. ⌘ n x n t f(x.u, t)=(1 t) = (x) n! n 0 X n and (x) = x(x + 1) (x + n 1) = (x + n 1)n. In particular we have < 1> ··· ⌘ (.u) . Solutions of (72) are therefore of the type [( 1.↵ + x.u).u]n s (x) . n ' n! Now we need to identify ↵. As in the previous example, the moments of such an umbra depend on the initial condition (73). Observe that, if sn(x)isaShe↵er n sequence with associated polynomials (x.u) /n!, then sn(x n)isaShe↵ersequence with associated polynomials [(x n).u]n x n + n 1 [(x 1).]n , n! ' n ' n! ✓ ◆ since [(x 1).]n (x 1) . Therefore due to Theorem 6.1 we have ' n [( 1.↵ +(x 1).u).]n (74) s (x n) . n ' n!

So the values of sn(1 n) in the initial condition (73) give exactly the moments of 1.↵. normalized byn!. Therefore, by observing that [( 1.↵ + x + n 1).]n n ( 1.↵.)k x + n 1 s (x) n ' n! ' k! n k kX=0 ✓ ◆ we have n x + n 1 s (x)= s (1 k) . n k n k kX=0 ✓ ◆ From a computational point of view, this formula is very easy to implement by using the recursion of the initial condition. If one would evaluate the moments of 128 Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato

1.↵. not by using the recursion of the initial condition, but with a closed form, some di↵erent considerations must be done. By using equivalence (74) we have n 1 n 1 n 1 [ 1.↵. +(n i 1).]i si(n 2i)= si(x i) x=n i | ' i! i=0 i=0 i=0 X X X and from the initial condition we have n 1 n j 1 ( 1.↵.)n ( 1.↵.)j [(n i j 1).]i n! ' j! i! ' j=0 i=0 (75) X X n 1 ( 1.↵. + ) ' (n 1)! where is an umbra such that k i k [(k i).] (76) k! k!k . ' i! ' i=0 X Observe that k k t k k (t.) k k t t k k! (t.) u (u.. ) ' (k t)! ' t ' D t=0 t=0 X X ✓ ◆ by using Corollary 7.2. The umbra is said to be the boolean cumulant umbra of D (cf. [3]). In particular the umbra has moments equal to the Fibonacci numbers, since has g.f. 1 1 f(,t)= = . 1 t(1 + t) 1 t t2 In terms of g.f.’s, equivalence (75) gives n n 1 ( 1.↵.) n ( 1.↵. + ) n 1 t 1+t t n! ' (n 1)! n 0 n 1 X X so that 1 f(,t) 1 =1+t f( 1.↵., t)= f(↵., t) f(↵., t) , 1 tf(,t) and 1.↵. u.. . ⌘ D Therefore the solution in closed form is [u.. +(x + n 1).]n s (x) D . n ' n! Example 7.8. Suppose we are asked to solve the di↵erence equation

(77) Fn(m)=Fn(m 1) + Fn 1(m 2) under the condition Fn(0) = 1 for all nonnegative integers n. Replace m with x + n + 1. Then equation (77) can be rewritten as

(78) Fn(x + n + 1) = Fn(x + n)+Fn 1(x + n 1) . Equation (78) ﬁts the She↵er identity (38) if we set y = 1, choose the sequence p (x) such that p (x)=1,p (1) = 1 and p (1) = 0 for all n 2 and consider { n } 0 1 n The classical umbral calculus: She↵er sequences 129 the She↵er sequence n!Fn(x) . As in Example 7.6, we are looking for solutions of (78) such that { } [( 1.↵ + x.u).]n F (x + n) . n ' n! Let us observe that equation (78), for x = 0, gives the well-known recurrence relation for Fibonacci numbers so that ( 1.↵.)n F (n) n . n ' n! ' n Therefore we have 1.↵. ,with n!n as given in equivalence (76), and ⌘ ' ↵ 1... Then, solutions of (78) are such that ⌘ ( + x.)n F (x + n) . n ' n! In particular n (x.)n k n (x.)n k k (j.)k j F (x + n) k n ' (n k)! ' (n k)! (k j)! ' j=0 kX=0 kX=0 X n n n k j j n n k (x.) (k.) [(x + k).] ' (n k j)! j! ' (n k)! ' j=0 kX=0 X kX=0 n x+k ' n k kX=0 ✓ ◆ by which we can verify that the initial conditions F (0) = F ( n + n) = 1 hold. n n References [1] A. Di Bucchianico & D. Loeb, A selected survey of umbral calculus,Electron.J.Combin., 2(1995), Dynamic Survey 3, pp. 28, (updated 2000). [2] A. Di Bucchianico & G. Soto y Koelemeijer, Solving linear recurrences using functionals, in Algebraic combinatorics and computer science,SpringerItalia,Milan,2001,461–472. [3] E. Di Nardo, P. Petrullo & D. Senato, Cumulants, convolutions and volume polynomials, Preprint, 2008. [4] E. Di Nardo & D. Senato, Umbral nature of the Poisson random variables,inAlgebraic combinatorics and computer science,SpringerItalia,Milan,2001,245–266. [5] E. Di Nardo & D. Senato, An umbral setting for cumulants and factorial moments,Eu- ropean J. Combin., 27(3)(2006) 394–413. [6] E. Di Nardo, G. Guarino & D. Senato, A unifying framework for k-statistics,polykays and their multivariate generalizations,Bernoulli,14(2)(2008),440-.468. [7] I.M. Gessel, Applications of the classical umbral calculus,AlgebraUniversalis, 49(4)(2003), 397–434. [8] H. Niederhausen, She↵er polynomials and linear recurrnces,CongressusNumerantium, 29(1980), 689–698. [9] H. Niederhausen, A formula for explicit solutions of certain linear recursions on polynomial sequences,CongressusNumerantium,49(1985),87–98. [10] H. Niederhausen, Recursive initial value problems for She↵er sequences,DiscreteMath., 204(1999), 319–327. [11] M. Razpet, An application of the umbral calculus,J.Math.Anal.Appl.,149(1990),1–16. [12] J. Riordan, An Introduction to combinatorial analysis, John Wiley and Sons, Inc. New York, 1958. [13] S.M. Roman, The umbral calculus,AcademicPress,NewYork,1984. [14] S.M. Roman & G.-C. Rota, The umbral calculus,Adv.Math.,27(1978),95–188. 130 Elvira Di Nardo, Heinrich Niederhausen and Domenico Senato

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