Combinatorial identities for Appell polynomials

Emanuele Munarini

doi:10.2298/aadm161001004m

Abstract

Using the techniques of the modern umbral calculus, we derive several combinatorial identities involving s-Appell polynomials. In particular, we obtain identities for classical polynomials, such as the Hermite, Laguerre, Bernoulli, Euler, N?rlund, hypergeometric Bernoulli, and Legendre polynomials. Moreover, we obtain a generalization of Carlitz's identity for Bernoulli numbers and polynomials to arbitrary symmetric s-Appell polynomials.

Highlights

An s-Appell polynomial sequence, with s = 0, is a polynomial sequence {pn(x)}n∈N generated by the formal exponential series p(x; t) =
A Sheffer sequence is a polynomial sequence {sn(x)}n∈N generated by the formal exponential series g(t) exf(t) n≥0 where g(t) =
In Lemmas we give umbral identities and in Propositions we give the corresponding identities for Appell polynomials

Summary

INTRODUCTION

An s-Appell polynomial sequence, with s = 0 , is a polynomial sequence {pn(x)}n∈N generated by the formal exponential series (1). 2k k!2n−2kxn−k k form a 2-Appell sequence, having generating exponential series tn e2xt. A Sheffer sequence is a polynomial sequence {sn(x)}n∈N generated by the formal exponential series (3). Given an s-Appell polynomial sequence {pn(x)}n∈N , with s = 0 , we define a linear isomorphism φ : R[x] → R[x] by setting φ(xn) = pn(x) , for all n ∈ N , and by extending it by linearity. This isomorphism has the following fundamental property.

A symmetric binomial identity

Symmetric Appell polynomials

SPECIAL IDENTITIES

Identities involving Stirling numbers