The Densities of Bounded Primes in Hypergeometric Series

Abstract

This thesis deals with the properties of the coefficients of Hypergeometric Series. Specifically, we are interested in which primes appear in the denominators to a bounded power. The first main result gives a method of categorizing the primes up to equivalence class which appear finitely many times in the denominators of generalized hypergeometric series nFm over the rational numbers. Necessary and sufficient conditions for when the density is zero are provided as well as a categorization of the n and m for which the problem is interesting.

The second main result is a similar condition for the appearance of primes in the denominators of the hypergeometric series 2F1 over number fields, specifically quadratic extensions Q(D). A novel conjecture to the study of p-adic numbers is also provided, which discusses the digits of irrational algebraic numbers' p-adic expansions.