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Some Supercongruences on q-Trinomial Coefficients

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Abstract

The trinomial coefficients \(\left( \!\!\!\genfrac(){0.0pt}0{n}{k}\!\!\!\right) \) are given by

$$\begin{aligned} \sum _{k=-n}^n\left( \!\!\!\genfrac(){0.0pt}0{n}{k}\!\!\!\right) x^k=(1+x+x^{-1})^n. \end{aligned}$$

Andrews and Baxter listed six kinds of q-trinomial coefficients (q-analogues of the trinomial coefficients). In this paper, we obtain some supercongruences on these q-trinomial coefficients. As a conclusion, we obtain the following new supercongruence:

$$\begin{aligned} \left( \!\!\!\left( {\begin{array}{c}ap\\ bp\end{array}}\right) \!\!\!\right) \equiv \left( \!\!\!\left( {\begin{array}{c}a\\ b\end{array}}\right) \!\!\!\right) \ (\mathrm{{mod}}\ p^2), \end{aligned}$$

where a, b are positive integers subject to \(a>b\) and \(p>3\) is an odd prime.

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Funding

The work is supported by the Natural Science Foundation of the Higher Education Institutions of Jiangsu Province (21KJB110001) and the National Natural Science Foundation of China (Grants 12001279 and 12201291).

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Authors and Affiliations

  1. Department of Applied Mathematics, Nanjing Audit University, Nanjing, 211815, People’s Republic of China

    He-Xia Ni

  2. School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing, 211816, People’s Republic of China

    Li-Yuan Wang

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  1. He-Xia Ni
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Correspondence to He-Xia Ni.

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Ni, HX., Wang, LY. Some Supercongruences on q-Trinomial Coefficients. Results Math 78, 130 (2023). https://doi.org/10.1007/s00025-023-01913-7

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