On unbounded denominators and hypergeometric series

TL;DR

This paper provides a criterion to determine when hypergeometric series coefficients are p-adically unbounded, characterizes the distribution of such primes, and explores connections to modular forms.

Contribution

It introduces a necessary and sufficient criterion for p-adic unboundedness of hypergeometric series coefficients applicable to almost all primes.

Findings

Set of unbounded primes is a finite union of arithmetic progressions.

The density of unbounded primes can be zero or one depending on parameters.

Explicit computation of unbounded prime sets is possible.

Abstract

We study the question of when the coefficients of a hypergeometric series are p-adically unbounded for a given rational prime p. Our first main result is a necessary and sufficient criterion (applicable to all but finitely many primes) for determining when the coefficients of a hypergeometric series with rational parameters are p-adically unbounded. This criterion is then used to show that the set of unbounded primes for a given series is, up to a finite discrepancy, a finite union of primes in arithmetic progressions. This set can be computed explicitly. We characterize when the density of the set of unbounded primes is 0, and when it is 1. Finally, we discuss the connection between this work and the unbounded denominators conjecture concerning Fourier coefficients of modular forms.