Orthogonality relations and generating functions for Jacobi polynomials and related hypergeometric functions

Abstract

The authors begin by examining the validity of some orthogonality relations and expansion formulas (asserted recently by S. D. Bajpai [1]) involving a class of hypergeometric polynomials which are essentially certain modified Jacobi polynomials. The corrected version of each of these orthogonality relations is shown to follow readily from the familiar orthogonality property of the classical Jacobi polynomials. A brief discussion is then presented about the applicability of an orthogonality property for the first few Jacobi polynomials, but over a semi-infinite interval, which was considered by V. Romanovski [2] and (more recently) by S. D. Bajpai [3]. Several families of generating functions for Jacobi and Laguerre polynomials, and for various related hypergeometric functions in one and more variables, are also considered systematically.