Skip to main content
SpringerLink
Log in

On two supercongruences for sums of Apéry-like numbers

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

We prove two supercongruences for sums of the Apéry-like numbers:

$$\begin{aligned} T_n=\sum _{k=0}^n{n\atopwithdelims ()k}^2{2k\atopwithdelims ()n}^2, \end{aligned}$$

which was first introduced by Almkvist and Zudilin. These results confirm two conjectural supercongruences due to Sun. Our proof relies on symbolic summation method and Sun’s transformation formula.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Almkvist, G., Zudilin, W.: Differential equations, mirror maps and zeta values. In: Symmetry, Mirror (ed.) V, AMS/IP Studies in Advanced Mathematics, pp. 481–515. American Mathematical Society, Providence (2006)

    Google Scholar 

  2. Amdeberhan, T., Tauraso, R.: A congruence for a double harmonic sum. Sci. Ser. A Math. Sci. (N.S.) 29, 37–44 (2019)

    MATH  Google Scholar 

  3. Apéry, R.: Irrationalité de \(\zeta (2)\) et \(\zeta (3)\). Astérisque 61, 11–13 (1979)

    MATH  Google Scholar 

  4. Chakraborty, K., Komatsu, T.: Generalized hypergeometric Bernoulli numbers. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 115, 101 (2021)

    Article  MathSciNet  Google Scholar 

  5. Guo, V.J.W.: \(q\)-Supercongruences modulo the fourth power of a cyclotomic polynomial via creative microscoping. Adv. Appl. Math. 120, 102078 (2020)

    Article  MathSciNet  Google Scholar 

  6. 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)

    Article  MathSciNet  Google Scholar 

  7. Guo, V.J.W., Liu, J.-C.: Some congruences related to a congruence of Van Hamme. Integral Transforms Spec. Funct. 31, 221–231 (2020)

    Article  MathSciNet  Google Scholar 

  8. Guo, V.J.W., Schlosser, M.J.: A new family of \(q\)-supercongruences modulo the fourth power of a cyclotomic polynomial. Results Math. 75, 155 (2020)

    Article  MathSciNet  Google Scholar 

  9. Guo, V.J.W., Zudilin, W.: A \(q\)-microscope for supercongruences. Adv. Math. 346, 329–358 (2019)

    Article  MathSciNet  Google Scholar 

  10. Kucukoglu, I., Simsek, Y.: On interpolation functions for the number of \(k\)-ary Lyndon words associated with the Apostol–Euler numbers and their applications. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 113, 281–297 (2019)

    Article  MathSciNet  Google Scholar 

  11. Li, L.: Some \(q\)-supercongruences for truncated forms of squares of basic hypergeometric series. J. Differ. Equ. Appl. 27, 16–25 (2021)

    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 Fís. Nat. Ser. A Mat. 114, 190 (2020)

    Article  MathSciNet  Google Scholar 

  13. Liu, J.-C.: Semi-automated proof of supercongruences on partial sums of hypergeometric series. J. Symb. Comput. 93, 221–229 (2019)

    Article  MathSciNet  Google Scholar 

  14. Liu, J.-C.: On Van Hamme’s (A.2) and (H.2) supercongruences. J. Math. Anal. Appl. 471, 613–622 (2019)

    Article  MathSciNet  Google Scholar 

  15. Liu, J.-C.: Some supercongruences arising from symbolic summation. J. Math. Anal. Appl. 488, 124062 (2020)

    Article  MathSciNet  Google Scholar 

  16. Liu, J.-C.: On a congruence involving \(q\)-Catalan numbers. C. R. Math. Acad. Sci. Paris 358, 211–215 (2020)

    MathSciNet  MATH  Google Scholar 

  17. Liu, J.-C.: On two congruences involving Franel numbers. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114, 201 (2020)

    Article  MathSciNet  Google Scholar 

  18. Liu, J.-C., Huang, Z.-Y.: A truncated identity of Euler and related \(q\)-congruences. Bull. Aust. Math. Soc. 102, 353–359 (2020)

    Article  MathSciNet  Google Scholar 

  19. Liu, J.-C., Petrov, F.: Congruences on sums of \(q\)-binomial coefficients. Adv. Appl. Math. 116, 102003 (2020)

    Article  MathSciNet  Google Scholar 

  20. Merca, M.: Bernoulli numbers and symmetric functions. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114, 20 (2020)

    Article  MathSciNet  Google Scholar 

  21. Schneider, C.: Symbolic summation assists combinatorics. Sém. Lothar. Combin. 56, B56b (2007)

    MathSciNet  MATH  Google Scholar 

  22. Shuang, Y., Guo, B.-N., Qi, F.: Logarithmic convexity and increasing property of the Bernoulli numbers and their ratios. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 115, 135 (2021)

    Article  MathSciNet  Google Scholar 

  23. Sun, Z.-H.: Congruences concerning Bernoulli numbers and Bernoulli polynomials. Discrete Appl. Math. 105, 193–223 (2000)

    Article  MathSciNet  Google Scholar 

  24. Sun, Z.-H.: Congruences involving Bernoulli and Euler numbers. J. Number Theory 128, 280–312 (2008)

    Article  MathSciNet  Google Scholar 

  25. Sun, Z.-H.: Super congruences for two Apéry-like sequences. J. Differ. Equ. Appl. 24, 1685–1713 (2018)

    Article  Google Scholar 

  26. Sun, Z.-W.: Super congruences and Euler numbers. Sci. China Math. 54, 2509–2535 (2011)

    Article  MathSciNet  Google Scholar 

  27. Sun, Z.-W.: New series for powers of \(\pi \) and related congruences. Electron. Res. Arch. 28, 1273–1342 (2020)

    Article  MathSciNet  Google Scholar 

  28. Wang, X., Yue, M.: Some \(q\)-supercongruences from Watson’s \(_8\phi _7\) transformation formula. Results Math. 75, 71 (2020)

    Article  Google Scholar 

  29. Wang, X., Yue, M.: A \(q\)-analogue of a Dwork-type supercongruence. Bull. Aust. Math. Soc. 103, 303–310 (2021)

    Article  MathSciNet  Google Scholar 

  30. Zhu, L.: New bounds for the ratio of two adjacent even-indexed Bernoulli numbers. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114, 83 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to thank the anonymous referees for their helpful comments which helped to improve the exposition of the paper. This work was supported by the National Natural Science Foundation of China (Grant 11801417).

Author information

Authors and Affiliations

  1. Department of Mathematics, Wenzhou University, Wenzhou, 325035, People’s Republic of China

    Ji-Cai Liu

Authors
  1. Ji-Cai Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ji-Cai Liu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, JC. On two supercongruences for sums of Apéry-like numbers. RACSAM 115, 151 (2021). https://doi.org/10.1007/s13398-021-01092-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-021-01092-6

Keywords

Mathematics Subject Classification

Search

Navigation