On the Atkin and Swinnerton-Dyer type congruences for some truncated hypergeometric ${}_1F_0$ series

TL;DR

This paper establishes Atkin and Swinnerton-Dyer type congruences for truncated hypergeometric series involving ${}_1F_0$ functions, revealing recursive relations modulo prime powers.

Contribution

It proves new congruences for truncated hypergeometric series, extending classical results to higher prime powers and specific parameters.

Findings

Derived congruences for sums involving hypergeometric series modulo p^{2α}.

Established recursive relations linking sums at different powers of p.

Extended Atkin and Swinnerton-Dyer type results to ${}_1F_0$ series.

Abstract

Let $p$ be an odd prime and let $n$ be a positive integer. For any positive integer $\alpha$ and $m\in\{1,2,3\}$, we have \begin{align*} \sum_{k=0}^{p^{\alpha}n-1}\frac{(\frac12)_k}{k!}\cdot\frac{(-4)^k}{m^k}\equiv\bigg(\frac{m(m-4)}{p}\bigg)\sum_{k=0}^{p^{\alpha-1}n-1}\frac{(\frac12)_k}{k!}\cdot\frac{(-4)^k}{m^k}\pmod{p^{2\alpha}}, \end{align*} where $(x)_k=x(x+1)\cdots(x+k-1)$ and $\big(\frac{\cdot}{\cdot}\big)$ denotes the Legendre symbol. Also, when $m=4$, \begin{align*} \sum_{k=0}^{p^{\alpha}n-1}(-1)^k\cdot\frac{(\frac12)_k}{k!}\equiv p\sum_{k=0}^{p^{\alpha-1}n-1}(-1)^k\cdot\frac{(\frac12)_k}{k!}\pmod{p^{2\alpha}}. \end{align*}