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Factors of some lacunary \(q\)-binomial sums

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Abstract

In this paper, we prove a divisibility result for the lacunary \(q\)-binomial sum

$$\begin{aligned} \sum _{k\equiv r\pmod {c}}(-1)^kq^{\left( {\begin{array}{c}k\\ 2\end{array}}\right) }{\genfrac[]{0.0pt}{}{n}{k}}_{q} {\genfrac[]{0.0pt}{}{(k-r)/c}{l}}_{q^{c}}. \end{aligned}$$

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References

  1. Andrews, G.E.: \(q\)-Analogs of the binomial coefficient congruences of Babbage, Wolstenholme and Glaisher. Discrete Math. 204, 15–25 (1999)

  2. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  3. Davis, D.M., Sun, Z.W.: A number-theoretic approach to homotopy exponents of \(SU(n)\). J. Pure Appl. Algebra 209, 57–69 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Désarménien, J.: Un analogue des congruences de Kummer pour les \(q\)-nombres d’Euler. Eur. J. Comb. 3, 19–28 (1982)

    Article  MATH  Google Scholar 

  5. Guo, V.J.W., Zeng, J.: Some arithmetic properties of the \(q\)-Euler numbers and \(q\)-Salié numbers. Eur. J. Comb. 27, 884–895 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Knuth, D., Wilf, H.: The power of a prime that divides a generalized binomial coefficient. J. Reine Angew. Math. 396, 212–219 (1989)

    MathSciNet  MATH  Google Scholar 

  7. Schultz, A., Walker, R.: A generalization of the Gaussian formula and a \(q\)-analog of Fleck’s congruence. J. Number. Theory. 133, 3717-3738 (2013)

    Google Scholar 

  8. Sun, Z.W.: Polynomial extension of Fleck’s congruence. Acta Arith. 122, 91–100 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Sun, Z.W.: Combinatorial congruences and Stirling numbers. Acta Arith. 126, 387–398 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Sun, Z.W., Davis, D.M.: Combinatorial congruences modulo prime powers. Trans. Am. Math. Soc. 359, 5525–5553 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wan, D.: Combinatorial congruences and \(\psi \)-operators. Finite Fields Appl. 12, 693–703 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Weisman, C.S.: Some congruences for binomial coefficients. Mich. Math. J. 24, 141–151 (1977)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author is supported by National Natural Science Foundation of China (Grant No. 10901078). I am grateful to the anonymous referee for his/her very helpful suggestions on this paper. I also thank Professor Zhi-Wei Sun for his useful discussions.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Hao Pan

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  1. Hao Pan
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Correspondence to Hao Pan.

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Communicated by C. Krattenthaler.

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Pan, H. Factors of some lacunary \(q\)-binomial sums. Monatsh Math 172, 387–398 (2013). https://doi.org/10.1007/s00605-013-0515-0

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  • DOI: https://doi.org/10.1007/s00605-013-0515-0

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