Abstract
Define \(g_n(x)=\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k\) for \(n=0,1,2,\ldots \). Those numbers \(g_n=g_n(1)\) are closely related to Apéry numbers and Franel numbers. In this paper we establish some fundamental congruences involving \(g_n(x)\). For example, for any prime \(p>5\) we have
This is similar to Wolstenholme’s classical congruences
for any prime \(p>3\).
Similar content being viewed by others
References
Andrews, G., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)
Barrucand, P.: A combinatorial identity, problem 75-4. SIAM Rev. 17, 168 (1975)
Callan, D.: A combinatorial interpretation for an identity of Barrucand. J. Integer Seq. 11, 3 (2008). Article 08.3.4
Gould, H.W.: Combinatorial Identities. Morgantown Printing and Binding Co., West Virginia (1972)
Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison-Wesley, New York (1994)
Guo, V.J.W.: Proof of two conjectures of Sun on congruences for Franel numbers. Integral Transform. Spec. Funct. 24, 532–539 (2013)
Guo, V.J.W., Zeng, J.: Proof of some conjectures of Z.-W. Sun on congruences for Apéry polynomials. J. Number Theory 132, 1731–1740 (2012)
Jarvis, F., Verrill, H.A.: Supercongruences for the Catalan–Larcombe–French numbers. Ramanujan J. 22, 171–186 (2010)
Petkovšek, M., Wilf, H.S., Zeilberger, D.: \(A=B\), A K Peters. Wellesley, Massachusetts (1996)
Sloane, N.J.A.: Sequence A000172 in OEIS (On-Line Encyclopedia of Integer Sequences). http://oeis.org/A000172
Strehl, V.: Recurrences and Legendre transform. Sém. Lothar. Comb. 29, 1–22 (1992)
Strehl, V.: Binomial identities—combinatorial and algorithmic aspects. Discret. Math. 136, 309–346 (1994)
Sun, Z.-W.: List of conjectural series for powers of \(\pi \) and other constants, preprint. arXiv:1102.5649
Sun, Z.-W.: On congruences related to central binomial coefficients. J. Number Theory 131, 2219–2238 (2011)
Sun, Z.-W.: Super congruences and Euler numbers. Sci. China Math. 54, 2509–2535 (2011)
Sun, Z.-W.: On sums of Apéry polynomials and related congruences. J. Number Theory 132, 2673–2699 (2012)
Sun, Z.-W.: Arithmetic theory of harmonic numbers. Proc. Am. Math. Soc. 140, 415–428 (2012)
Sun, Z.-W.: On sums of binomial coefficients modulo \(p^2\). Colloq. Math. 127, 39–54 (2012)
Sun, Z.-W.: Connections between \(p=x^2+3y^2\) and Franel numbers. J. Number Theory 133, 2914–2928 (2013)
Sun, Z.-W.: Congruences for Franel numbers. Adv. Appl. Math. 51, 524–535 (2013)
Sun, Z.-W.: p-adic congruences motivated by series. J. Number Theory 134, 181–196 (2014)
Sun, Z.-W., Tauraso, R.: New congruences for central binomial coefficients. Adv. Appl. Math. 45, 125–148 (2010)
Sun, Z.-W., Tauraso, R.: On some new congruences for binomial coefficients. Int. J. Number Theory 7, 645–662 (2011)
Tauraso, R.: More congruences for central binomial congruences. J. Number Theory 130, 2639–2649 (2010)
van der Poorten, A.: A proof that Euler missed \(\ldots \) Asperys proof of the irrationality of \(\zeta (3)\). Math. Intelligencer 1, 195–203 (1978/79)
Wolstenholme, J.: On certain properties of prime numbers. Quart. J. Appl. Math. 5, 35–39 (1862)
Zagier, D.: Integral solutions of Apéry-like recurrence equations. In: Groups and Symmetries: From Neolithic Scots to John McKay. CRM Proceedings of the Lecture Notes 47. American Mathematics Society, Providence, pp. 349–366 (2009)
Zhao, J.: Wolstenholme type theorem for multiple harmonic sums. Int. J. Number Theory 4, 73–106 (2008)
Acknowledgments
The author would like to thank the referee for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Science Foundation (Grant No. 11571162) of China.
Rights and permissions
About this article
Cite this article
Sun, ZW. Congruences involving \(g_n(x)=\sum \limits _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) ^2\left( {\begin{array}{c}2k\\ k\end{array}}\right) x^k\) . Ramanujan J 40, 511–533 (2016). https://doi.org/10.1007/s11139-015-9727-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-015-9727-3