On Jacobi and continuous Hahn polynomials

Abstract

Jacobi polynomials are mapped onto the continuous Hahn polynomials by the Fourier transform and the orthogonality relations for the continuous Hahn polynomials then follow from the orthogonality relations for the Jacobi polynomials and the Parseval formula. In a special case this relation dates back to work by Bateman in 1933 and we follow a part of the historical development for these polynomials. Some applications of this relation are given. 1. Introduction and history In Askey's scheme of hypergeometric orthogonal polynomials we find the Jacobi polyno- mials and the continuous Hahn polynomials; see Askey and Wilson (5, Appendix) with the correction in (2), Koekoek and Swarttouw (20) or Koornwinder (23, §5) for information on Askey's scheme. In the hierarchy of Askey's scheme of hypergeometric orthogonal polyno- mials the continuous Hahn polynomials are above the Jacobi polynomials since they have one extra degree of freedom. In this paper we consider a way of going up in the Askey scheme from the Jacobi polynomials to the continuous Hahn polynomials by use of the Fourier transform. This method is a simple extension of some special cases introduced by Bateman in the 1930's. In his 1933 paper (9) Bateman introduces the polynomial Fn satisfying