Extended Laguerre Polynomials Associated with Hermite, Bernoulli, and Euler Numbers and Polynomials

Abstract

Let ${\mathbf{P}}_n=\{p(x)\in ℝ[x]\mid \mathrm{deg}\hspace{0.17em}\hspace{0.17em}p(x)\le n\}$ be an inner product space with the inner product $\langle p(x),q(x)\rangle ={\int }_0^{\infty }{x}^{\alpha }{e}^{-x}p(x)q(x)dx$, where $p(x),q(x)\in {\mathbf{P}}_n$ and $\alpha \in ℝ$ with $\alpha >-1$. In this paper we study the properties of the extended Laguerre polynomials which are an orthogonal basis for ${\mathbf{P}}_n$. From those properties, we derive some interesting relations and identities of the extended Laguerre polynomials associated with Hermite, Bernoulli, and Euler numbers and polynomials.