Properties of multiple Hermite and multiple Laguerre polynomials by the generating function

Abstract

A generating function is obtained for multiple Hermite and multiple Laguerre polynomials from the Rodrigues formula, and then interesting recurrence relations are given. We also find a relation between multiple orthogonal polynomials and ordinary orthogonal polynomials such as multiple Hermite polynomials with classical Hermite polynomials and multiple Laguerre polynomials with classical Laguerre polynomials. Lastly, we find a differential equation for the multiple orthogonal polynomials by the generating function, which is a reconstruction of the works by Aptekarev et al. [Aptekarev, A.I., Branquinho, A., Van Assche, W., 2003, Multiple orthogonal polynomials for classical weights. Transaction of American Mathematical Society, 355, 3887–3914.].