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

Abstract

This article draws on results from [Triković, 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 [Triković, 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.