The action of Hecke operators on hypergeometric functions

TL;DR

This paper investigates how Hecke operators act on hypergeometric functions and formal power series, revealing their spectral properties and the significance of polylogarithms in eigenfunction analysis.

Contribution

It characterizes the spectrum of Hecke operators on hypergeometric functions and highlights the role of polylogarithms in eigenfunction structure.

Findings

Spectrum of Hecke operators is n^a with integer a and positive integer n.

Polylogarithms are central to the eigenfunctions of these operators.

Results imply new insights into hypergeometric coefficients and their multiplicative properties.

Abstract

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.