Biorthogonal polynomials suggested by the jacobi polynomials

Abstract

In this paper we introduce and study a pair of biorthogonal polynomials that are suggested by the classical Jacobi polynomials. Let α> − l,β> − l and jn(α, β, k; x) and Kn(χ, β, k; x), n=0, 1, 2, be respectively the polynomials of degree n in xk and x, where x is real, k is a positive integer such that these two polynomial sets satisfy biorthogonality conditions with respect to the weight function (1−x)α(l+x)β, namely and conversely. For k=l both these sets are reduced to the Jacobi polynomial sets. We obtain generating functions, recurrence relations for both these sets and explicitlyshow that they satisfy biorthogonality conditions. © 1982, University of California, Berkeley. All Rights Reserved.