The Action of Hecke Operators on Hypergeometric Functions

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 corollary of our results on simultaneous eigenfunctions, we also obtain an apriori unrelated result regarding the behavior of completely multiplicative hypergeometric coefficients.