Algorithms for Algebraic and Arithmetic Attributes of Hypergeometric Functions

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.

💡 Research Summary

The paper “Algorithms for Algebraic and Arithmetic Attributes of Hypergeometric Functions” by Xavier Caruso and Florian Fürnsinn presents a comprehensive algorithmic framework for studying hypergeometric series with rational parameters from both p‑adic and modular perspectives. The authors restrict themselves to series H(α,β;x)=∑_{k≥0} (α₁)_k…(α_n)_k/(β₁)_k…(β_m)_k·x^k that lie in Q