Weights whose biorthogonal polynomials admit a Rodrigues formula

Abstract

Let α > 0 and ψ ( x ) = x α . Let w be a non-negative integrable function on an interval I. Let P n be a polynomial of degree n determined by the biorthogonality conditions ∫ I P n ψ j w = 0 , j = 0 , 1 , … , n − 1 . We determine for which weights w, P n admits an analogue of the classical Rodrigues formula for orthogonal polynomials, and present the formula whenever it exists. We also provide generating functions and fairly explicit representations for P n .