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On Some Conjectural Supercongruences for Sums Involving Certain Rising Factorials

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Abstract

We here deduce some supercongruence results for certain sums involving rising factorials using a method similar to the WZ-method. As particular cases, we confirm certain recent conjectural supercongruence of Guo.

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References

  1. Cohen, H.: Number Theory. Volume II: Analytic and Modern Tools, vol. 240. Springer, New York (2008)

  2. Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications, vol. 96, 2nd edn. Cambridge University Press, Cambridge (2004)

  3. Gu, C.-Y., Guo, V.J.W.: Two \(q\)-congruences from Carlitz’s formula. Period. Math. Hung. 82, 82–86 (2021)

    Article  MathSciNet  Google Scholar 

  4. Guo, V.J.W.: Some generalizations of a supercongruence of van Hamme. Integral Transforms Spec. Funct. 28(12), 888–899 (2017)

    Article  MathSciNet  Google Scholar 

  5. Guo, V.J.W., Schlosser, M.J.: Some new \(q\)-congruences for truncated basic hypergeometric series: even powers. Results Math. 75(1), 1–15 (2020)

    Article  MathSciNet  Google Scholar 

  6. Guo, V.J.W., Schlosser, M.J.: A family of \(q\)-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial. Israel J. Math. 240, 821–835 (2020)

    Article  MathSciNet  Google Scholar 

  7. Guo, V.J.W., Schlosser, M.J.: Proof of a basic hypergeometric supercongruence modulo the fifth power of a cyclotomic polynomial. J. Differ. Equ. Appl. 25, 921–929 (2019)

    Article  MathSciNet  Google Scholar 

  8. Guo, V.J.W., Wang, S.-D.: Some congruences involving fourth powers of central \(q\)-binomial coefficients. Proc. Roy. Soc. Edinburgh Sect. A 150, 1127–1138 (2020)

    Article  MathSciNet  Google Scholar 

  9. Jana, A., Kalita, G.: Supercongruences for sums involving rising factorial \(\left(\frac{1}{\ell }\right)_k^3\). Integral Transforms Spec. Funct. 30(9), 683–692 (2019)

    Article  MathSciNet  Google Scholar 

  10. Jana, A., Kalita, G.: Supercongruences for sums involving fourth power of some rising factorials. Proc. Math. Sci. 130(1), 1–13 (2020)

    Article  MathSciNet  Google Scholar 

  11. Kalita, G., Jana, A.: On some supercongruence conjectures for truncated hypergeometric series. Indian J. Pure Appl. Math (2021). https://doi.org/10.1007/s13226-021-00072-1

    Article  MathSciNet  Google Scholar 

  12. Li, L., Wang, S.D.: Proof of a \(q\)-supercongruence conjectured by Guo and Schlosser. Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser. A Mat. 114(4), 1–7 (2020)

  13. Long, L.: Hypergeometric evaluation identities and supercongruences. Pac. J. Math. 249(2), 405–418 (2011)

    Article  MathSciNet  Google Scholar 

  14. Long, L., Ramakrishna, R.: Some supercongruences occuring in truncated hypergeometric series. Adv. Math. 290, 773–808 (2016)

    Article  MathSciNet  Google Scholar 

  15. Mortenson, E.: A \(p\)-adic supercongruence conjecture of vanHamme. Proc. Am. Math. Soc. 136(12), 4321–4328 (2008)

    Article  MathSciNet  Google Scholar 

  16. Petkov\(\hat{s}\)ek, M., Wilf, H.S., Zeilberger, D.: \(A =B\), A. K. Peters, Ltd., Wellesley, Mass. (1996)

  17. Ramanujan, S.: Modular equations and approximations to\(\pi \), Quart. J. Math. 45, (1914), 350-372. In Collected papers of Srinivasa Ramanujan, pages 23–39. AMS Chelsea Publ., Providence, RI, (2000)

  18. Sun, Z.-W.: Supercongruences motivated by \(e\). J. Number Theory 147, 326–341 (2015)

    Article  MathSciNet  Google Scholar 

  19. Sun, Z.-W.: Binomial coefficients, Catalan numbers and Lucas quotients. Sci. China Math. 53, 2473–2488 (2010)

    Article  MathSciNet  Google Scholar 

  20. VanHamme, L.: Some conjectures concerning partial sums of generalized hypergeometric series, \(p\)-adic functional analysis. Lect. Notes Pure Appl. Math. 192(1997), 223–236 (1996)

    Google Scholar 

  21. Wang, C., Ni, H.-X.: Some q-congruences arising from certain identities, arXiv:2003.10883, (Preprint)

  22. Wang, X., Yue, M.: Some \(q\)-supercongruences from Watson’s \(_8\phi _7\) transformation formula. Results Math. 75, 1–5 (2020)

    Article  Google Scholar 

  23. Zudilin, W.: Ramanujan-type supercongruences. J. Number Theory 129(8), 1848–1857 (2009)

    Article  MathSciNet  Google Scholar 

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Funding

The first author acknowledges the support received from the Department of Science and Technology, Government of India, through an INSPIRE Fellowship DST/INSPIRE Fellowship/2017/IF170327. The second author is partially supported by a project EMR/2016/005010 of SERB, Department of Science and Technology, Government of India, under Extra Mural Research Funding (Individual Centric).

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Correspondence to Gautam Kalita.

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Jana, A., Kalita, G. On Some Conjectural Supercongruences for Sums Involving Certain Rising Factorials. Results Math 76, 155 (2021). https://doi.org/10.1007/s00025-021-01469-4

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