An extended class of orthogonal polynomials defined by a Sturm-Liouville problem

Abstract

We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as X(1)-Jacobi and X(1)-Laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval [-1, 1] or the half-line [0, infinity), respectively, and they are a basis of the corresponding L(2) Hilbert spaces. Moreover, we prove a converse statement similar to Bochner's theorem for the classical orthogonal polynomial systems: if a self-adjoint second-order operator has a complete set of polynomial eigenfunctions {p(i)}(i=1)(infinity), then it must be either the X(1)-Jacobi or the X(1)-Laguerre Sturm-Liouville problem. A Rodrigues-type formula can be derived for both of the X(1) polynomial sequences.