Quadratic sums of Gaussian q-binomial coefficients and Fibonomial coefficients

  • Wenchang Chu
  • Emrah Kılıç


In this paper we evaluate quadratic sums of Gaussian q-binomial coefficients with two additional parameters. We obtain a general summation theorem using a combination of Heine’s transformation, the q-Pfaff–Saalschutz theorem and the q-Kummer sum. Consequently several identities for generalized Fibonomial–Lucanomial coefficients are obtained by specifying the parameter p and the base q.


Basic hypergeometric series q-Binomial coefficient Heine transformation q-Pfaff–Saalschutz summation formula q-Kummer sum Fibonomial and Lucanomial coefficients 


  1. 1.
    Chen, X., Chu, W.: Summation formulae for a class of terminating balanced \(q\)-series. J. Math. Anal. Appl. 451(1), 508–523 (2017)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Gasper, G., Rahman, M.: Basic Hypergeometric Series, 2nd edn. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  3. 3.
    Gould, H.W.: The bracket function and Fountené-Ward generalized binomial coefficients with application to fibonomial coefficients. Fibonacci Q. 7, 23–40 (1969)MATHGoogle Scholar
  4. 4.
    Hoggatt Jr., V.E.: Fibonacci numbers and generalized binomial coefficients. Fibonacci Q. 5, 383–400 (1967)MathSciNetMATHGoogle Scholar
  5. 5.
    Kılıç, E., Prodinger, H.: Evaluation of sums involving Gaussian \(q\)-binomial coefficients with rational weight functions. Int. J. Number Theory 12(2), 495–504 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kılıç, E., Prodinger, H.: Closed form evaluation of sums containing squares of Fibonomial coefficients. Math. Slovaca 66(3), 757–765 (2016)MathSciNetMATHGoogle Scholar
  7. 7.
    Kılıç, E., Prodinger, H.: Closed form evaluation of restricted sums containing squares of Fibonomial coefficients. U.P.B. Sci. Bull. Ser. A 78(4), 57–66 (2016)MathSciNetMATHGoogle Scholar
  8. 8.
    Kılıç, E., Prodinger, H., Akkuş, I., Ohtsuka, H.: Formulas for Fibonomial sums with generalized Fibonacci and Lucas coefficients. Fibonacci Q. 49(4), 320–329 (2011)MathSciNetMATHGoogle Scholar
  9. 9.
    Kılıç, E., Ohtsuka, H., Akkuş, I.: Some generalized Fibonomial sums related with the Gaussian \(q\)-binomial sums. Bull. Math. Soc. Sci. Math. Roum. 55(1), 51–61 (2012)MathSciNetMATHGoogle Scholar
  10. 10.
    Li, N.N., Chu, W.: \(q\)-Derivative operator proof for a conjecture of Melham. Discret. Appl. Math. 177, 158–164 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Seibert, J., Trojovsky, P.: On some identities for the Fibonomial coefficients. Math. Slovaca 55, 9–19 (2005)MathSciNetMATHGoogle Scholar
  12. 12.
    Trojovsky, P.: On some identities for the Fibonomial coefficients via generating function. Discret. Appl. Math. 155(15), 2017–2024 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhoukou Normal UniversityZhoukouPeople’s Republic of China
  2. 2.Mathematics DepartmentTOBB University of Economics and TechnologySögütözüTurkey
  3. 3.Dipartimento di Matematica e Fisica “Ennio De Giorgi”Università del SalentoLecceItaly

Personalised recommendations