Q-orthogonal polynomials

Abstract

This chapter and the next one have many things in common. The generating function technique by Rainville is used to prove recurrences for q-Laguerre polynomials. We prove product expansions and bilinear generating functions for q-Laguerre polynomials by using operator formulas. Many formulas for q-Laguerre polynomials are special cases of q-Jacobi polynomial formulas. A certain Rodriguez operator turns up, a generalization of the Rodriguez formula. We will prove orthogonality for both the above-mentioned polynomials by using q-integration by parts, a method equivalent to recurrences. Many of the operator formulas for q-Jacobi polynomial formulas are formal, because of the limited convergence region of the q-shifted factorial. The q-Legendre polynomials are defined by the Rodrigues formula to enable an easy orthogonality relation. q-Legendre polynomials have been given before, but these do not have the same orthogonality range in the limit as ordinary Legendre polynomials. We also find q-difference equations for these polynomials.