We study the action of the Hecke operators Un on the set of hypergeometric functions, as well as on formal power series. We show that the spectrum of these operators on the set of hypergeometric functions is the set {na : n ∈ N and a ∈ Z}, and that the polylogarithms play a dominant role in the study of the eigenfunctions of the Hecke operators Un on the set of hypergeometric functions. As a co...

In many problems arising in physical sciences and statistics, hypergeometric functions, in logarithmic cases, are applicable, which is known from the monograph of Mathai and Haubold [4] and Mathai [3], etc. A detailed account of such applications is available from the two monographs by Mathai and Saxena [5], [6]. Expansions of the Gauss’s hypergeometric functions in logarithmic cases are given ...