Density formulas for $p$-adically bounded primes for hypergeometric series with rational and quadratic irrational parameters

TL;DR

This paper investigates the distribution of primes for which hypergeometric series with rational or quadratic irrational parameters are p-adically bounded, providing exact formulas and bounds for their densities.

Contribution

It extends density formulas for hypergeometric series with rational parameters and establishes bounds and conjectures for quadratic irrational parameters.

Findings

Exact density formula for rational parameters with finite monodromy.

Unconditional lower bound on density for quadratic irrational parameters.

Examples of series attaining the upper bound of 1/2, mainly associated with imaginary quadratic fields.

Abstract

We study densities of $p$-adically bounded primes for hypergeometric series in two cases: the case of generalized hypergeometric series with rational parameters, and the case of $_2F_1$ with parameters in a quadratic extension of the rational numbers. In the rational case we extend work from $_2F_1$ to $_nF_{n-1}$ for an exact formula giving the density of bounded primes for the series. The density is shown to be one exactly in accordance with the case of finite monodromy as classified by Beukers-Heckmann. In the quadratic irrational case, we obtain an unconditional lower bound on the density of bounded primes. Assuming the normality of the $p$-adic digits of quadratic irrationalities, this lower bound is shown to be an exact formula for the density of bounded primes. In the quadratic irrational case, there is a trivial upper bound of $1/2$ on the density of bounded primes. In the final section of the paper we discuss some results and computations on series that attain this bound. In particular, all such examples we have found are associated to imaginary quadratic fields, though we do not prove this is always the case.