Differential equation derived from generating function – the case of disk polynomials

Abstract

There are a variety of characterizations in classical orthogonal polynomials. First of all, by explicit expressions, secondly by generating functions, thirdly as polynomial solutions of differential equations. There also exist other ways. Needless to say, these definitions are equivalent one another.

In the case of the disk polynomials, a similar situation occurs. However, there are few studies that refer to relations with their generating function. The purpose of this paper is to show by a heuristic method that the partial differential equations of second order which the disk polynomials satisfy are derived from their generating function.