Extended Laguerre polynomials associated with Hermite, Bernoulli, and Euler numbers and polynomials (Q1925468)

Summary: Let \(\mathbf P_n = \{p(x) \in \mathbb R[x]\mid \deg p(x) \leq n\}\) be an inner product space with the inner product \(\langle p(x), q(x)\rangle = \int^\infty_0 x^\alpha e^{-x} p(x)q(x)\,dx\), where \(p(x), q(x) \in\mathbf P_n\) and \(\alpha \in \mathbb R\) 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.