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PROOF OF TWO CONJECTURES ON SUPERCONGRUENCES INVOLVING CENTRAL BINOMIAL COEFFICIENTS

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

Abstract

In this note we use some $q$ -congruences proved by the method of ‘creative microscoping’ to prove two conjectures on supercongruences involving central binomial coefficients. In particular, we confirm the $m=5$ case of Conjecture 1.1 of Guo [‘Some generalizations of a supercongruence of Van Hamme’, Integral Transforms Spec. Funct.28 (2017), 888–899].

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The second author was partially supported by the National Natural Science Foundation of China (Grant No. 11771175).

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References

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[1]Gu, C.-Y. and Guo, V. J. W., ‘q-Analogues of two supercongruences of Z.-W. Sun’, Czechoslovak Math. J. to appear.
[2]Guo, V. J. W., ‘Some generalizations of a supercongruence of van Hamme’, Integral Transforms Spec. Funct. 28 (2017), 888899.
[3]Guo, V. J. W., ‘Common q-analogues of some different supercongruences’, Results Math. 74 (2019), Article ID 131.
[4]Guo, V. J. W. and Liu, J.-C., ‘Some congruences related to a congruence of Van Hamme’, Integral Transforms Spec. Funct. to appear.
[5]Guo, V. J. W. and Schlosser, M. J., ‘Some new q-congruences for truncated basic hypergeometric series: even powers’, Results Math. 75 (2020), Article ID 1.
[6]Guo, V. J. W. and Zudilin, W., ‘A q-microscope for supercongruences’, Adv. Math. 346 (2019), 329358.
[7]Hou, Q.-H., Mu, Y.-P. and Zeilberger, D., ‘Polynomial reduction and supercongruences’, J. Symbolic Comput. to appear.
[8]Liu, J.-C., ‘Semi-automated proof of supercongruences on partial sums of hypergeometric series’, J. Symbolic Comput. 93 (2019), 221229.
[9]Mortenson, E., ‘A p-adic supercongruence conjecture of van Hamme’, Proc. Amer. Math. Soc. 136 (2008), 43214328.
[10]Straub, A., ‘Supercongruences for polynomial analogs of the Apéry numbers’, Proc. Amer. Math. Soc. 147 (2019), 10231036.
[11]Sun, Z.-W., ‘A refinement of a congruence result by van Hamme and Mortenson’, Illinois J. Math. 56 (2012), 967979.
[12]Swisher, H., ‘On the supercongruence conjectures of van Hamme’, Res. Math. Sci. 2 (2015), Article ID 18.
[13]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.
[14]Zudilin, W., ‘Ramanujan-type supercongruences’, J. Number Theory 129 (2009), 18481857.
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PROOF OF TWO CONJECTURES ON SUPERCONGRUENCES INVOLVING CENTRAL BINOMIAL COEFFICIENTS

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

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