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Some q-supercongruences from Watson’s \(_8\phi _7\) Transformation Formula

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Abstract

Recently, Jana and Kalita proved the following supercongruences involving rising factorials \((\frac{1}{d})_k^3\):

$$\begin{aligned}&\sum _{k=0}^{N} (-1)^k (2dk+1)\frac{(\frac{1}{d})_k^3}{k!^3}\\&\quad \equiv {\left\{ \begin{array}{ll} p^r&{}\pmod {p^{r+2}},\quad \text {if } r \text { is even};\\ (-1)^{\frac{(p-d+1)r}{d}}(d-1)p^r&{}\pmod {p^{r+2}},\quad \text {if } r \text { is odd}; \end{array}\right. }\\&\sum _{k=0}^{N} (-1)^k (2dk+1)^3\frac{(\frac{1}{d})_k^3}{k!^3}\\&\quad \equiv {\left\{ \begin{array}{ll} -3p^r&{}\pmod {p^{r+2}},\quad \text {if } r \text { is even};\\ (-1)^{\frac{(p+1)r}{d}}3(d-1)p^r&{}\pmod {p^{r+2}},\quad \text {if } r \text { is odd}, \end{array}\right. } \end{aligned}$$

where \(N={\left\{ \begin{array}{ll} \frac{p^r-1}{d}, \quad &{}\text {if } r \text { is even};\\ \frac{(d-1)p^r-1}{d},\quad &{}\text {if } r \text { is odd}.\end{array}\right. }\) From Watson’s \(_8\phi _7\) transformation formula, we give q-analogues of the above supercongruences, generalizing some previous conjectural results of Van Hamme. Our proof uses the ‘creative microscoping’ method which was introduced by Guo and Zudilin.

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  1. Department of Applied Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

    Xiaoxia Wang & Mingbing Yue

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  1. Xiaoxia Wang
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Correspondence to Mingbing Yue.

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This work is supported by National Natural Science Foundations of China (11661032).

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Wang, X., Yue, M. Some q-supercongruences from Watson’s \(_8\phi _7\) Transformation Formula. Results Math 75, 71 (2020). https://doi.org/10.1007/s00025-020-01195-3

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