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

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