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A $q$ -ANALOGUE OF A HYPERGEOMETRIC CONGRUENCE

  • CHENG-YANG GU (a1) and VICTOR J. W. GUO (a2)

Abstract

We give a $q$ -analogue of the following congruence: for any odd prime $p$ ,

$$\begin{eqnarray}\mathop{\sum }_{k=0}^{(p-1)/2}(-1)^{k}(6k+1)\frac{(\frac{1}{2})_{k}^{3}}{k!^{3}8^{k}}\mathop{\sum }_{j=1}^{k}\biggl(\frac{1}{(2j-1)^{2}}-\frac{1}{16j^{2}}\biggr)\equiv 0\;(\text{mod}\;p),\end{eqnarray}$$
which was originally conjectured by Long and later proved by Swisher. This confirms a conjecture of the second author [‘A $q$ -analogue of the (L.2) supercongruence of Van Hamme’, J. Math. Anal. Appl. 466 (2018), 749–761].

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The second author was partially supported by the National Natural Science Foundation of China (grant 11771175).

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References

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[1] Gosper, W., ‘Strip mining in the abandoned orefields of nineteenth century mathematics’, in: Computers in Mathematics (eds. Chudnovsky, D. V. and Jenks, R. D.) (Dekker, New York, 1990), 261284.
[2] Guo, V. J. W., ‘A q-analogue of the (L.2) supercongruence of Van Hamme’, J. Math. Anal. Appl. 466 (2018), 749761.
[3] Guo, V. J. W., ‘A q-analogue of a curious supercongruence of Guillera and Zudilin’, J. Difference Equ. Appl. 25 (2019), 342350.
[4] Guo, V. J. W., ‘Factors of some truncated basic hypergeometric series’, J. Math. Anal. Appl. 476 (2019), 851859.
[5] Guo, V. J. W. and Schlosser, M. J., ‘Some new q-congruences for truncated basic hypergeometric series’, Symmetry 11(2) (2019), Article 268, 12 pages.
[6] Guo, V. J. W. and Zudilin, W., ‘Ramanujan-type formulae for 1/𝜋: q-analogues’, Integral Transforms Spec. Funct. 29 (2018), 505513.
[7] Guo, V. J. W. and Zudilin, W., ‘A q-microscope for supercongruences’, Adv. Math. 346 (2019), 329358.
[8] Long, L., ‘Hypergeometric evaluation identities and supercongruences’, Pacific J. Math. 249 (2011), 405418.
[9] Osburn, R. and Zudilin, W., ‘On the (K.2) supercongruence of Van Hamme’, J. Math. Anal. Appl. 433 (2016), 706711.
[10] Rahman, M., ‘Some quadratic and cubic summation formulas for basic hypergeometric series’, Canad. J. Math. 45 (1993), 394411.
[11] Ramanujan, S., ‘Modular equations and approximations to 𝜋’, Quart. J. Math. Oxford Ser. (2) 45 (1914), 350372.
[12] Straub, A., ‘Supercongruences for polynomial analogs of the Apéry numbers’, Proc. Amer. Math. Soc. 147 (2019), 10231036.
[13] Swisher, H., ‘On the supercongruence conjectures of van Hamme’, Res. Math. Sci. 2 (2015), Article 18, 21 pages.
[14] Van Hamme, L., ‘Some conjectures concerning partial sums of generalized hypergeometric series’, in: p-Adic Functional Analysis (Nijmegen, 1996), Lecture Notes in Pure and Applied Mathematics, 192 (Dekker, New York, 1997), 223236.
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