On the orthogonality of classical orthogonal polynomials

Abstract

We consider the orthogonality of rational functions Wn(s) as the Laplace transform images of a set of orthoexponential functions, obtained from the Jacobi polynomials, and as the Laplace transform images of the Laguerre polynomials. We prove that the orthogonality of the Jacobi and the Laguerre polynomials is induced by the orthogonality of the functions Wn(s). Thus we have shown that the orthogonality relations of the Jacobi and Laguerre polynomials are equivalent to the orthogonality of rational functions which are essentially the images of the classical orthogonal polynomials under the Laplace transform.