Some properties of q-biorthogonal polynomials

Abstract

Almost four decades ago, Konhauser introduced and studied a pair of biorthogonal polynomials Y α n (x; k) and Z α n (x; k) α > −1; k ∈ N := {1, 2, 3, . . .} , which are suggested by the classical Laguerre polynomials. The so-called Konhauser biorthogonal polynomials Z α n (x; k) of the second kind were indeed considered earlier by Toscano without their biorthogonality property which was emphasized upon in Konhauser's investigation. Many properties and results for each of these biorthogonal polynomials (such as generating functions, Rodrigues formulas, recurrence relations, and so on) have since been obtained in several works by others. The main object of this paper is to present a systematic investigation of the general family of q-biorthogonal polynomials. Several interesting properties and results for the q-Konhauser polynomials are also derived.