Abstract
We prove the divisibility conjecture on sums of even powers of q-binomial coefficients, which was recently proposed by Guo, Schlosser and Zudilin. Our proof relies on two q-harmonic series congruences due to Shi and Pan.
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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)
Gorodetsky, O.: \(q\)-Congruences, with applications to supercongruences and the cyclic sieving phenomenon. Int. J. Number Theory 15, 1919–1968 (2019)
Gu, C.-Y., Guo, V.J.W.: \(q\)-Analogues of two supercongruences of Z.-W. Sun. Czech. Math. J. 70, 757–765 (2020)
Guo, V.J.W.: A \(q\)-analogue of the (A.2) supercongruence of Van Hamme for primes \(p\equiv 1~(mod \; 4)\). Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114, Art. 123 (2020)
Guo, V.J.W.: Proof of a generalization of the (C.2) supercongruence of Van Hamme. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 115, Art. 45 (2021)
Guo, V.J.W., Liu, J.-C.: \(q\)-Analogues of two Ramanujan-type formulas for \(1/\pi \). J. Differ. Equ. Appl. 24, 1368–1373 (2018)
Guo, V.J.W., Schlosser, M.J.: Some \(q\)-supercongruences modulo the square and cube of a cyclotomic polynomial. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 115, Art. 132 (2021)
Guo, V.J.W., Schlosser, M.J., Zudilin, W.: New quadratic identities for basic hypergeometric series and \(q\)-congruences, preprint (2021). http://math.ecnu.edu.cn/~jwguo/maths/quad.pdf
Guo, V.J.W., Zeng, J.: New congruences for sums involving Apéry numbers or central Delannoy numbers. Int. J. Number Theory 8, 2003–2016 (2012)
Guo, V.J.W., Zudilin, W.: A \(q\)-microscope for supercongruences. Adv. Math. 346, 329–358 (2019)
Li, L., Wang, S.-D.: Proof of a \(q\)-supercongruence conjectured by Guo and Schlosser. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114, Art. 190 (2020)
Liu, J.-C.: On a congruence involving \(q\)-Catalan numbers. C. R. Math. Acad. Sci. Paris 358, 211–215 (2020)
Liu, J.-C.: A variation of the \(q\)-Wolstenholme theorem. Ann. Mat. Pura Appl. https://doi.org/10.1007/s10231-022-01187-w(online)
Liu, J.-C.: On the divisibility of \(q\)-trinomial coefficients. Ramanujan J. https://doi.org/10.1007/s11139-022-00558-4(online)
Liu, J.-C., Huang, Z.-Y.: A truncated identity of Euler and related \(q\)-congruences. Bull. Aust. Math. Soc. 102, 353–359 (2020)
Liu, J.-C., Petrov, F.: Congruences on sums of \(q\)-binomial coefficients. Adv. Appl. Math. 116, 102003 (2020)
Pan, H.: A generalization of Wolstenholme’s harmonic series congruence. Rocky Mt. J. Math. 38, 1263–1269 (2008)
Shi, L.-L., Pan, H.: A \(q\)-analogue of Wolstenholme’s harmonic series congruence. Am. Math. Mon. 114, 529–531 (2007)
Wang, C., Ni, H.-X.: Some \(q\)-congruences arising from certain identities. Period. Math. Hungar. https://doi.org/10.1007/s10998-021-00416-8(online)
Wang, X., Yu, M.: Some new \(q\)-congruences on double sums. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 115, Art. 9 (2021)
Wei, C.: Some \(q\)-supercongruences modulo the fourth power of a cyclotomic polynomial. J. Combin. Theory Ser. A 182, 105469 (2021)
Acknowledgements
The authors would like to thank the anonymous referees for their helpful comments which helped to improve the exposition of the paper. The first author was supported by the National Natural Science Foundation of China (grant 12171370).
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Liu, JC., Jiang, XT. On the divisibility of sums of even powers of q-binomial coefficients. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116, 76 (2022). https://doi.org/10.1007/s13398-022-01220-w
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DOI: https://doi.org/10.1007/s13398-022-01220-w