On the adelic Gaussian hypergeometric function

TL;DR

This paper introduces an adelic hypergeometric function constructed via hypergeometric curves, which unifies finite field hypergeometric functions and relates to adelic beta functions, with proven transformation and summation formulas.

Contribution

It defines a new adelic hypergeometric function that interpolates classical hypergeometric functions over all finite fields and establishes key transformation and summation formulas.

Findings

Defines the adelic hypergeometric function using hypergeometric curves

Shows the function interpolates all finite field hypergeometric functions

Proves classical transformation and summation formulas in the adelic setting

Abstract

We define the adelic hypergeometric function of special Gaussian type by means of a tower of hypergeometric curves. This function takes values in an adelic completed group ring and interpolates all the hypergeometric functions of the same type over all finite fields. It specializes at the unit argument to the adelic beta function of Ihara and Anderson. We prove some transformation formulas and a summation formula for the adelic hypergeometric function, which are known classically for complex hypergeometric functions.