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## Acta Arithmetica

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

### Volume 198 / 2021

Acta Arithmetica 198 (2021), 169-186 MSC: Primary 05A10, 11A07, 33C20; Secondary 11B39, 11B65, 11B75. DOI: 10.4064/aa200405-8-8 Published online: 30 December 2020

#### Abstract

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, $$\sum _{k=0}^{p^{r}n-1}\frac {\left (\frac 12\right )_k}{k!}\equiv p\sum _{k=0}^{p^{r-1}n-1}\frac {\left (\frac 12\right )_k}{k!}\pmod {p^{2r}}.$$

#### Authors

• Yong ZhangDepartment of Mathematics and Physics
Nanjing Institute of Technology
Nanjing 211167, People’s Republic of China
e-mail
• Hao PanSchool of Applied Mathematics
Nanjing University
of Finance and Economics
Nanjing 210023, People’s Republic of China
e-mail

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