Extended laguerre polynomials associated with hermite, bernoulli, and euler numbers and polynomials

Abstract

Let P n = {p(x)∈ ℝ[x] | deg p(x) ≤ n} be an inner product space with the inner product 〈 p(x), q(x) 〉 = ∫ 0∞ x α e -x p(x)q(x)dx, where p(x), q(x) ∈ P n and α ∈ ℝwith α > -1. In this paper we study the properties of the extended Laguerre polynomials which are an orthogonal basis for P n. From those properties, we derive some interesting relations and identities of the extended Laguerre polynomials associated with Hermite, Bernoulli, and Euler numbers and polynomials. © 2012 Taekyun Kim and Dae San Kim.