The action of Hecke operators on hypergeometric functions

TL;DR

This paper investigates how Hecke operators act on hypergeometric functions, revealing their spectral properties and the role of polylogarithms, leading to a characterization of hypergeometric coefficients.

Contribution

It provides a new analysis of Hecke operators on hypergeometric functions, identifying their spectrum and the significance of polylogarithms in eigenfunctions.

Findings

Spectrum consists of powers n^a

Polylogarithms are central to eigenfunctions

Characterization of multiplicative hypergeometric coefficients

Abstract

We study the action of Hecke operators on the set of hypergeometric functions. We show that the spectrum of these operators is the set of powers n^a and that polylogarithms play a dominant role in the study of the corresponding eigenfunctions. As a corollary, we obtain a characterization of completely multiplicative hypergeometric coefficients.