Generalized Power Series Expansions for a Class of Orthogonal Polynomials in Two Variables

This paper continues the analysis of a class of orthogonal polynomials in two variables on a region bounded by two straight lines and a parabola touching these lines, which was introduced by the first author. An explicit series expansion for these polynomials is obtained, which generalizes Constantine’s expansion of hypergeometric functions of (2 × 2) matrix argument in terms of James’ zonal polynomials. In two special cases the orthogonal polynomials turn out to be Appell’s hypergeometric F₄-functions and certain hypergeometric functions in two variables of order three, respectively.