The Bateman Functions Revisited after 90 Years—A Survey of Old and New Results

Abstract

The Bateman functions and the allied Havelock functions were introduced as solutions of some problems in hydrodynamics about ninety years ago, but after a period of one or two decades they were practically neglected. In handbooks, the Bateman function is only mentioned as a particular case of the confluent hypergeometric function. In order to revive our knowledge on these functions, their basic properties (recurrence functional and differential relations, series, integrals and the Laplace transforms) are presented. Some new results are also included. Special attention is directed to the Bateman and Havelock functions with integer orders, to generalizations of these functions and to the Bateman-integral function known in the literature.

FIVE SERIES EQUATIONS INVOLVING GENERALIZED BATEMAN k-FUNCTIONS

“…Recently Tripathi and Dixit [20] have obtained formal solution of four series equations involving generalized Bateman k-functions. Apelblat, Consiglia and Mainardi [1] in a recent survey expressed that Havlock(1925) and Bateman(1931) has introduced new functions as solutions of fluid dynamics problems. Recently, Shrivastava, Narain and Shrivastava [17] obtained solution of triple series equations involving generalized Laguerre polynomials.…”