On the Atkin and Swinnerton-Dyer type congruences for some truncated hypergeometric ${}_1F_0$ series
Tom 198 / 2021
Acta Arithmetica 198 (2021), 169-186
MSC: Primary 05A10, 11A07, 33C20; Secondary 11B39, 11B65, 11B75.
DOI: 10.4064/aa200405-8-8
Opublikowany online: 30 December 2020
Streszczenie
Let $p$ be an odd prime and let $n$ be a positive integer with $p\nmid n$. For any positive integer $r$ and $\lambda \in \{1, 2, 3\}$ with $p\nmid \lambda $, we have $$ \sum _{k=0}^{p^{r}n-1}\frac {\left (\frac 12\right )_k}{k!}\cdot \frac {4^k}{\lambda ^k}\equiv \bigg (\frac {\lambda (\lambda -4)}{p}\bigg )\sum _{k=0}^{p^{r-1}n-1}\frac {\left (\frac 12\right )_k}{k!}\cdot \frac {4^k}{\lambda ^k}\pmod {p^{2r}}, $$ where $(x)_k=x(x+1)\cdots (x+k-1)$ and $\big(\frac{\cdot}{\cdot}\big) $ denotes the Legendre symbol. Also, $$ \sum _{k=0}^{p^{r}n-1}\frac {\left (\frac 12\right )_k}{k!}\equiv p\sum _{k=0}^{p^{r-1}n-1}\frac {\left (\frac 12\right )_k}{k!}\pmod {p^{2r}}. $$
