A new approach to the orthogonality of the Laguerre and Hermite polynomials

Abstract

This article draws on results from [Tričković, S.B. and Stanković, M.S., 2003, On the orthogonality of classical orthogonal polynomials. Integral Transforms and Special Functions, 14(3), 271–280.], where we considered the orthogonality of rational functions W n (s) which are obtained as the images of the classical orthogonal polynomials under the Laplace transform. We proved in [Tričković, S.B. and Stanković, M.S., 2003, On the orthogonality of classical orthogonal polynomials. International Transaction of Specific Function, 14(3), 271–280.] that the orthogonality relations of the Jacobi polynomials and the standard Laguerre polynomials L n (x) are induced by and are equivalent to the orthogonality of rational functions W n (s). In this article, we continue in the same manner by considering the generalized Laguerre polynomials and Hermite polynomials H n (x). In the last section, we analyze the zeros distribution of the Laplace transform images of the Legendre, Chebyshev, Laguerre and Hermite polynomials.