Evaluation of sums of products of Gaussian q-binomial coefficients with rational weight functions

Generalizing earlier results, sums over the products of the Gaussian q-binomial coefficients are computed. Some applications of the results for special choices of q are emphasized. The results are obtained by the elementary technique of partial fraction decomposition.

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  • [1] Andrews GE, Askey R, Roy R. Special Functions. Cambridge, UK: Cambridge University Press, 2000.
  • [2] Cao J, Chu W. Two transformations for a binomial sum. Mathematical Communications 2016; 21 (2): 219-225.
  • [3] Chu W. Partial–fraction decompositions and harmonic number identities. Journal of Combinatorial Mathematics and Combinatorial Computing 2007; 60: 139-153.
  • [4] Chu W. Partial fraction decompositions and trigonometric sum identities. Proceedings of the American Mathematical Society 2008; 136 (1): 229-237.
  • [5] Egorychev GP. Integral Representation and the Computation of Combinatorial Sums. Providence, RI, USA: American Mathematical Society Translations of Mathematical Monographs, 1984.
  • [6] Gasper G. Rahman M. Basic Hypergeometric Series, Second Ed. Cambridge, UK: Cambridge University Press, 2004.
  • [7] Kılıç E, Akkuş İ, Prodinger H. A proof of a conjecture of Melham. Fibonacci Quarterly 2010; 48 (3): 241-248.
  • [8] Kılıç E, Arıkan T. 103.26 A proof of Clarke’s conjecture. The Mathematical Gazette 2019; 103: 346-352. 317
  • [9] Kılıç E, Ohtsuka H, Akkuş İ. Some generalized Fibonomial sums related with the Gaussian q -binomial sums. Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie (N.S.) 2012; 55 (103): 51-61.
  • [10] Kılıç E, Prodinger H. Evaluation of sums involving Gaussian q -binomial coefficients with rational weight functions. International Journal of Number Theory 2016; 12 (2): 495-504.
  • [11] Kılıç E, Prodinger H. Evaluation of sums involving products of Gaussian q -binomial coefficients with applications to Fibonomial sums. Turkish Journal of Mathematics 2017; 41: 707-716.
  • [12] Li NN, Chu W. q -derivative operator proof for a conjecture of Melham. Discrete Applied Mathematics 2014; 177: 158-164.
  • [13] Marques D, Trojovsky P. On some new sums of Fibonomial coefficients. Fibonacci Quarterly 2012; 50 (2): 155-162.
  • [14] Melham RS. Families of identities involving sums of powers of the Fibonacci and Lucas numbers. Fibonacci Quarterly 1999; 37 (4): 315-319.
  • [15] Prodinger H. Mortenson’s identities and partial fraction decomposition. Utilitas Mathematica 2017; 103: 175-179.
  • [16] Seibert J, Trojovsky P. On some identities for the Fibonomial coefficients. Mathematica Slovaca 2005; 55 (1): 9-19.
  • [17] Trojovsky P. On some identities for the Fibonomial coefficients via generating function. Discrete Applied Mathematics 2007; 155 (15): 2017-2024.