Algorithms for Algebraic and Arithmetic Attributes of Hypergeometric Functions

TL;DR

This paper presents algorithms to analyze the arithmetic properties of hypergeometric functions, including p-adic valuations, prime reduction sets, and annihilating polynomials, advancing computational methods in number theory.

Contribution

It introduces new algorithms for computing p-adic valuations, prime reduction sets, and annihilating polynomials of hypergeometric functions, extending previous work by Christol.

Findings

Computed p-adic valuations on disks within the radius of convergence

Determined prime sets for reduction of hypergeometric functions

Developed an algorithm to find annihilating polynomials modulo p

Abstract

We discuss algorithms for arithmetic properties of hypergeometric functions. Most notably, we are able to compute the p-adic valuation of a hypergeometric function on any disk of radius smaller than the p-adic radius of convergence. This we use, building on work of Christol, to determine the set of prime numbers modulo which it can be reduced. Moreover, we describe an algorithm to find an annihilating polynomial of the reduction of a hypergeometric function modulo p.