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Some new q-congruences on double sums

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

Swisher confirmed an interesting congruence: for any odd prime p,

$$\begin{aligned} \sum _{k=0}^{(p-1)/2}(-1)^k(6k+1)\frac{(\frac{1}{2})_k^{3}}{k!^{3}8^k} \sum _{j=1}^{k}\Bigg (\frac{1}{(2j-1)^{2}}-\frac{1}{16j^{2}}\Bigg ) \equiv 0 \pmod {p}, \end{aligned}$$

which was conjectured by Long. Recently, its q-analogue was proved by Gu and Guo. Inspired by their work, we obtain a new similar q-congruence modulo \(\Phi _n(q)\) and two q-supercongruences modulo \(\Phi _n(q)^{2}\) on double basic hypergeometric sums, where \(\Phi _n(q)\) is the n-th cyclotomic polynomial.

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Authors and Affiliations

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

    Xiaoxia Wang & Menglin Yu

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  1. Xiaoxia Wang
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  2. Menglin Yu
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Correspondence to Menglin Yu.

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

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Wang, X., Yu, M. Some new q-congruences on double sums. RACSAM 115, 9 (2021). https://doi.org/10.1007/s13398-020-00946-9

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