Abstract
Recently, Wei, Liu, and Wang gave a q-analogue of a supercongruence of Long and Ramakrishna (Adv Math 290:773–808, 2016) from the q-Pfaff–Saalschütz identity. In this note, we present a generalization of Wei–Liu–Wang’s q-supercongruence with one more parameter by using the q-Pfaff–Saalschütz identity and the method of ‘creative microscoping’ introduced by the second author and Zudilin again.
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References
El Bachraoui, M.: On supercongruences for truncated sums of squares of basic hypergeometric series. Ramanujan J. 54, 415–426 (2021)
Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and its Applications, vol. 96, 2nd edn. Cambridge University Press, Cambridge (2004)
Guo, V.J.W.: Factors of some truncated basic hypergeometric series. J. Math. Anal. Appl. 476, 851–859 (2019)
Guo, V.J.W.: A new extension of the (A.2) supercongruence of Van Hamme. Results Math. 77, 96 (2022)
Guo, V.J.W., Schlosser, M.J.: A family of \(q\)-supercongruences modulo the cube of a cyclotomic polynomial. Bull. Aust. Math. Soc. 105, 296–302 (2022)
Guo, V.J.W., Zudilin, W.: A \(q\)-microscope for supercongruences. Adv. Math. 346, 329–358 (2019)
Liu, Y., Wang, X.: Some \(q\)-supercongruences from a quadratic transformation by Rahman. Results Math. 77, 44 (2022)
Liu, J.-C.: A variation of the \(q\)-Wolstenholme theorem. Ann. Mat. Pura Appl. 201, 1993–2000 (2022)
Liu, J.-C., Jiang, X.-T.: On the divisibility of sums of even powers of \(q\)-binomial coefficients. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 116, 76 (2022)
Liu, J.-C., Petrov, F.: Congruences on sums of \(q\)-binomial coefficients. Adv. Appl. Math. 116, 102003 (2020)
Long, L., Ramakrishna, R.: Some supercongruences occurring in truncated hypergeometric series. Adv. Math. 290, 773–808 (2016)
Robert, A.M.: A Course in \(p\)-Adic Analysis. Graduate Texts in Mathematics, Springer, New York (2000)
Wei, C.: Some \(q\)-supercongruences modulo the fourth power of a cyclotomic polynomial. J. Combin. Theory Ser. A 182, 105469 (2021)
Wei, C., Liu, Y., Wang, X.: \(q\)-Supercongruences from the \(q\)-Saalschütz identity. Proc. Am. Math. Soc. 149, 4853–4861 (2021)
Xu, C., Wang, X.: Proofs of Guo and Schlosser’s two conjectures. Period. Math. Hung. (2022). https://doi.org/10.1007/s10998-022-00452-y
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The first author was partially supported by the National Natural Science Foundation of China (Grant No. 12271200).
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Gu, CY., Guo, V.J.W. Further q-Supercongruences from the q-Pfaff–Saalschütz Identity. Results Math 77, 238 (2022). https://doi.org/10.1007/s00025-022-01772-8
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DOI: https://doi.org/10.1007/s00025-022-01772-8