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

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.