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

Abstract

Various interesting and potentially useful properties and relationships involving the Bernoulli, Euler and Genocchi polynomials have been investigated in the literature rather extensively. Recently, the present authors (Srivastava and Pinter in Appl Math Lett 17:375–380, 2004) obtained addition theorems and other relationships involving the generalized Bernoulli polynomials \({B_n^{(\alpha)}(x)}\) and the generalized Euler polynomials \({E_n^{(\alpha)}(x)}\) 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 \({B_n^{(\alpha)}(x)}\) and the generalized Euler polynomials \({E_n^{(\alpha)}(x)}\) , but also the generalized Genocchi polynomials \({G_n^{(\alpha)}(x)}\) , the Srivastava polynomials \({\mathcal{S}_{n}^{N}\left( x\right)}\) , 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.