Addition theorems for the Appell polynomials and the associated classes of polynomial expansions

Abstract

n (x) and the generalized Euler polynomials E (α) n (x) of order α and degree n in x. The main purpose of this sequel to some of the aforecited investigations is to give several addition formulas for a general class of Appell sequences. The addition formulas, which are derived in this paper, involve not only the generalized Bernoulli polynomials B (α) n (x) and the generalized Euler polynomials E (α) n (x), but also the generalized Genocchi polynomials G (α) n (x), the Srivastava polynomials S N n (x), several general families of hypergeometric polynomials and such orthogonal polynomials as the Jacobi, Laguerre and Hermite polynomials. Some umbral-calculus generalizations of the addition formulas are also investigated.Mathematics Subject Classification (2010). Primary 11B68; Secondary 11B83, 33C45.

A novel approach for obtaining new identities for the lambda extension of q-Euler polynomials arising from the q-umbral calculus

“…Recent investigations on this topic include an elementary and real approach to values of the Riemann zeta function [5], Apostol-Euler polynomials arising from the umbral calculus [16], a new generalization of q-Bernoulli polynomial [17], a new q-generalization of Euler numbers and polynomials using the method of Kupershmidt [11], additional theorems for the Appell polynomials and associated classes of polynomial expansions [21], theorems on Apostol-Euler polynomials arising from Euler basis [27], a new class of q-Euler and q-Bernoulli polynomials [19,20], a modified q-Euler numbers of higher order with weight [22], the zeta and the q-zeta functions and associated series and integrals [29].…”

Some properties of Hermite based Appell matrix polynomials

“…Recently Srivastava et al [5,7,8,15,17,18] has established some results for the sheffer and Appell polynomials and also for the new classes of mixed special polynomials related to these polynomials by employing certain operational methods. This paper is an attempt to further stress the importance of operational methods in introducing new families of special matrix polynomials.…”

Srivastava-Pintér theorems for 2D-Appell polynomials and their applications

“…The probabilistic proofs of the Srivastava–Pintér addition theorems were given recently in . On the other hand, some addition theorems for the Appell polynomials and the associated classes of polynomial expansions were given in . Because the aforementioned polynomials are the representatives of the Appell polynomials, the polynomial set which are generated by the previous results motivate us to ask the following question:…”

“….˛, 2 C, n 2 N 0 /. Appell polynomials and the associated classes of polynomial expansions were given in [12]. Because the aforementioned polynomials are the representatives of the Appell polynomials, the polynomial set which are generated by…”