Splitting Hypergeometric Functions over Roots of Unity

TL;DR

This paper introduces formulas that decompose hypergeometric functions over roots of unity across finite field, p-adic, and classical contexts, leading to new reduction, summation, and evaluation formulas, and revealing connections to modular forms.

Contribution

It provides novel splitting formulas for hypergeometric functions over roots of unity in multiple mathematical settings, enabling new computational and theoretical insights.

Findings

New reduction and summation formulas for finite field hypergeometric functions

Explicit evaluations of special values in finite field and p-adic contexts

Discovery of relations between hypergeometric functions and modular form Fourier coefficients

Abstract

We examine hypergeometric functions in the finite field, p-adic and classical settings. In each setting, we prove a formula which splits the hypergeometric function into a sum of lower order functions whose arguments differ by roots of unity. We provide multiple applications of these results, including new reduction and summation formulas for finite field hypergeometric functions, along with classical analogues; evaluations of special values of these functions which apply in both the finite field and p-adic settings; and new relations to Fourier coefficients of modular forms.