# On integrality properties of hypergeometric series

## TL;DR

This paper characterizes when hypergeometric series solutions have p-integral coefficients based on the rationality and p-integrality of their parameters, advancing understanding of their integrality properties.

## Contribution

It provides a characterization of parameters leading to p-integral hypergeometric series coefficients, extending previous integrality results.

## Key findings

- Identifies conditions for p-integrality of hypergeometric series coefficients.
- Establishes criteria for rational and p-integral parameters within [-1,0].
- Derives new integrality results for hypergeometric series.

## Abstract

Let $A$ be a set of $N$ vectors in ${\mathbb Z}^n$ and let $v$ be a vector in ${\mathbb C}^N$ that has minimal negative support for $A$. Such a vector $v$ gives rise to a formal series solution of the $A$-hypergeometric system with parameter $\beta=Av$. If $v$ lies in ${\mathbb Q}^n$, then this series has rational coefficients. Let $p$ be a prime number. We characterize those $v$ whose coordinates are rational, $p$-integral, and lie in the closed interval $[-1,0]$ for which the corresponding normalized series solution has $p$-integral coefficients. From this we deduce further integrality results for hypergeometric series.