  We study the action of the Hecke operators Un on the set of hy- pergeometric
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 n^a with a an
integer and n a positive integer, 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 eigen-
functions, we also obtain an apriori unrelated result regarding the behavior of
completely multiplicative hypergeometric coefficients.
