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.
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Calkin, N.J.: A curious binomial identity. Discrete Math. 131, 335–337 (1994)
Graham, R.L.; Knuth, D.E.; Patashnik, O.: Concrete Mathematics a Foundation for Computer Science. Addison-Wesley, Boston (1992)
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)
He, B.: Some identities involving the partial sum of \(q\)-binomial coefficients. Electron. J. Comb. 21(3), P3.17 (2014)
Hirschhorn, M.: Calkin’s binomial identity. Discrete Math. 159, 273–278 (1996)
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)
Kılıç, E.; Prodinger, H.: Some Gaussian binomial sum formulæ with applications. Indian J. Pure Appl. Math. 47(3), 399–407 (2016)
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)
Mansour, T.; Shattuck, M.: A \(q\)-analog of the hyperharmonic numbers. Afr. Math. 25, 147–160 (2014)
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)
Ollerton, R.L.: Partial row-sums of Pascal’s triangle. Int. J. Math. Educ. Sci. Technol. 38(1), 124–127 (2005)
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