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Evaluation of sums involving products of Gaussian q -binomial coefficients with applications to Fibonomial sums

Sums of products of two Gaussian q -binomial coefficients with a parametric rational weight function are considered. The partial fraction decomposition technique is used to evaluate the sums in closed form. Interesting applications of these results to certain generalized Fibonomial and Lucanomial sums are provided. | Turk J Math (2017) 41: 707 – 716 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1503-57 Research Article Evaluation of sums involving products of Gaussian q -binomial coefficients with applications to Fibonomial sums 1 Emrah KILIC ¸ 1,∗, Helmut PRODINGER2 Department of Mathematics, TOBB Economics and Technology University, Ankara, Turkey 2 Department of Mathematics, University of Stellenbosch, Stellenbosch, South Africa • Received: 17.03.2015 Accepted/Published Online: 08.07.2016 • Final Version: 22.05.2017 Abstract: Sums of products of two Gaussian q -binomial coefficients with a parametric rational weight function are considered. The partial fraction decomposition technique is used to evaluate the sums in closed form. Interesting applications of these results to certain generalized Fibonomial and Lucanomial sums are provided. Key words: Gaussian q -binomial coefficients, Fibonomial coefficients, Lucanomial coefficients, sum identities 1. Introduction Define the second-order linear sequences {Un } and {Vn } for n ≥ 2 by Un = pUn−1 + Un−2 , Vn = pVn−1 + Vn−2 , U0 = 0, U1 = 1, V0 = 2, V1 = p. The Binet forms are Un = αn − β n 1 − qn = αn−1 α−β 1−q and Vn = αn + β n = αn (1 + q n ) √ with q = β/α = −α−2 , so that α = i/ q . When α = √ 1+ 5 2 √ √ (or equivalently q = (1− 5 )/(1+ 5 ) ), the sequence {Un } is reduced to the Fibonacci sequence {Fn } and the sequence {Vn } is reduced to the Lucas sequence {Ln } . Throughout this paper we will use the following notations: the q -Pochhammer symbol (x; q)n = (1 − x)(1 − xq) . . . (1 − xq n−1 ) and the Gaussian q -binomial coefficients as [ ] n (z; q)n = . k z (z; q)k (z; q)n−k When z = q, we denote (q; q)n by (q)n . Furthermore, we will use generalized Fibonomial coefficients { } n Un Un−1 . . . Un−k+1 = k U U 1 U 2 . . . Uk ∗Correspondence: ekilic@etu.edu.tr 2010 AMS Mathematics Subject Classification: Primary 11B39; Secondary 05A30. 707 KILIC ¸ and .