Biorthogonal polynomials suggested by the laguerre polynomials

Abstract

Let Y n(x;k) and Z(x;k), n = 0,1,, be polynomials of degree n in x and xk, respectively, where x is real, & is a positive integer and c > -1, such that For, conditions (1) and (2) reduce to the orthogonality requirement satisfied by the generalized Laguerre polynomials. If (1) and (2) hold, then and conversely. For both sets of polynomials, we shall establish mixed recurrence relations from which we shall derive differential equations of order k + 1. From these mixed recurrence relations pure recurrence relations connecting k + 2 successive polynomials can also be obtained. For k = 1, the recurrence relations and the differential equations for both of the polynomial sets reduce to those for the generalized Laguerre polynomials. © 1967 by Pacific Journal of Mathematics.