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Arabian Journal of Mathematics

, Volume 7, Issue 2, pp 101–112 | Cite as

Evaluation of various partial sums of Gaussian q-binomial sums

  • Emrah Kılıç
Open Access
Article
  • 160 Downloads

Abstract

We present three new sets of weighted partial sums of the Gaussian q-binomial coefficients. To prove the claimed results, we will use q-analysis, Rothe’s formula and a q-version of the celebrated algorithm of Zeilberger. Finally we give some applications of our results to generalized Fibonomial sums.

Mathematics Subject Classification

11B65 05A30 

Notes

References

  1. 1.
    Calkin, N.J.: A curious binomial identity. Discrete Math. 131, 335–337 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Graham, R.L.; Knuth, D.E.; Patashnik, O.: Concrete Mathematics a Foundation for Computer Science. Addison-Wesley, Boston (1992)zbMATHGoogle Scholar
  3. 3.
    Guo, V.J.W.; Lin, Y.-J.; Liu, Y.; Zhang, C.: A \(q\)-analogue of Zhang’s binomial coefficient identities. Discrete Math. 309, 5913–5919 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    He, B.: Some identities involving the partial sum of \(q\)-binomial coefficients. Electron. J. Comb. 21(3), P3.17 (2014)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hirschhorn, M.: Calkin’s binomial identity. Discrete Math. 159, 273–278 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kılıç, E.; Yalçıner, A.: New sums identities in weighted Catalan triangle with the powers of generalized Fibonacci and Lucas numbers. Ars Comb. 115, 391–400 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Kılıç, E.; Prodinger, H.: Some Gaussian binomial sum formulæ with applications. Indian J. Pure Appl. Math. 47(3), 399–407 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kılıç, E.; Prodinger, H.: Evaluation of sums involving products of Gaussian \(q\)-binomial coefficients with applications to Fibonomial sums. Turk. J. Math. 41(3), 707–716 (2017)MathSciNetGoogle Scholar
  9. 9.
    Mansour, T.; Shattuck, M.: A \(q\)-analog of the hyperharmonic numbers. Afr. Math. 25, 147–160 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mansour, T.; Shattuck, M.; Song, C.: \(q\)-Analogs of identities involving harmonic numbers and binomial coefficients. Appl. Appl. Math. Int. J. 7(1), 22–36 (2012)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ollerton, R.L.: Partial row-sums of Pascal’s triangle. Int. J. Math. Educ. Sci. Technol. 38(1), 124–127 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Mathematics DepartmentTOBB Economics and Technology UniversityAnkaraTurkey

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