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The Ramanujan Journal

, Volume 48, Issue 1, pp 173–189 | Cite as

Finding more divisibility properties of binomial sums via the WZ method

  • Brian Y. SunEmail author
Article
  • 92 Downloads

Abstract

In recent years, we have witnessed numerous new results on divisibility properties concerning binomial sums. Many conjectures and results were proposed by Z.-W. Sun. Some of these conjectures were proved by himself, and also by some scholars such as V.J.W. Guo, G.S. Mao, B. He, and Y.P. Mu. Motivated by their work, we study the WZ method and its applications to prove divisibility properties of binomial sums. In this paper, we propose a method which can be used to explore more such divisibility properties. Additionally, we give some generalized divisibility properties which can imply those divisibility properties established by Z.-W. Sun and B. He.

Keywords

Hypergeometric function Divisibility Binomial coefficients Binomial sums The WZ method 

Mathematics Subject Classification

11A05 11A07 05A10 11B65 

Notes

Acknowledgements

We are extremely thankful to the referee for helpful suggestions and comments, which greatly helped to improve the presentation of this paper.

References

  1. 1.
    Bober, J.W.: Factorial ratios, hypergeometric series, and a family of step functions. J. Lond. Math. Soc. 79, 422–444 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Calkin, N.J.: Factors of sums of powers of binomial coefficients. Acta Arith. 86, 17–26 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, Y.G., Xie, X.Y., He, B.: On some congruences of certain binomial sums. Ramanujan J. 40(2), 237–244 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gosper, R.W.: Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75, 40–42 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Guo, V.J.W., Krattenthaler, C.: Some divisibility properties of binomial and $q$-binomial coefficients. J. Number Theory 135, 167–184 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Guo, V.J.W.: Proof of Sun’s conjectures on integer-valued polynomials. J. Math. Anal. Appl. 444, 182–191 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Guo, V.J.W., Jouhet, F., Zeng, J.: Factors of alternating sums of products of binomial and $q$-binomial coefficients. Acta Arith. 127, 17–31 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Guo, V.J.W., Zeng, J.: Factors of binomial sums from the Catalan triangle. J. Number Theory. 130, 172–186 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    He, B.: On the divisibility properties of certain binomial sums. J. Number Theory. 147, 133–140 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    He, B.: On the divisibility properties concerning sums of binomial coefficients. Ramanujan J. 43(2), 313–326 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Petkovšek, M., Wilf, H.S., Zeilberger, D.: A = B. A K Peters, Wellesley (1996)CrossRefzbMATHGoogle Scholar
  12. 12.
    Sun, Z.-W.: Super congruences and Euler numbers. Sci. China Math. 54, 2509–2535 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Sun, Z.-W.: On divisibility of binomial coefficients. J. Aust. Math. Soc. 93, 189–201 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sun, Z.-W.: Products and sums divisible by central binomial coefficients. Electron. J. Comb. 20(1), P9 (2013)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Sun, B.Y.: On some divisibility properties of binomial sums. Int. J. Number Theory. 13(9), 2265–2276 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sun, Z.-W.: Open conjectures on congruences. arXiv:0911.5665
  17. 17.
    Warnaar, S.O., Zudilin, W.: A $q$-rious positivity. Aequ. Math. 81, 177–183 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Wilf, H.S., Zeilberger, D.: Rational functions certify combinatorial identities. J. Am. Math. Soc. 3, 147–158 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zudilin, W.: Ramanujan-type supercongruences. J. Number Theory. 129, 1848–1857 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and System ScienceXinjiang UniversityUrumqiPeople’s Republic of China

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