Some Special Functions

Abstract

Three types of special functions are considered: (1) Cylindrical functions, (2) Jacobi functions (this further subdivides into two subcases), and (3) Confluent hypergeometric functions. Some properties of these functions, which are needed in studying the expansions in terms of them, are discussed. In particular, various differentiation formulas, integral representations, and asymptotic estimates are presented. The key formulas are the Koornwinder integral representations for Jacobi functions and their analogues for the Kummer confluent hypergeometric function. These formulas generalize the classical Mehler–Dirichlet representation for Legendre functions and allow one to obtain Paley–Wiener-type theorems for the corresponding integral transforms.