On Binomial Sums and Alternating Binomial Sums of Generalized Fibonacci Numbers with Falling Factorials
In this paper, we consider and obtain binomial sums and alternating binomial sums including falling factorial of the summation indice. For example, For integers n and m such that 0 ≤ m < n, Xn k=0 n k k mU 2m 2k = n m (p 2 + 4)m Xm i=0 (−1)i 2m i V n−m 2(m−i) V2(m+n)(m−i) − (−1)m 2 n−m 2m m ! , and for positive odd integer m, Xn k=0 (−1)k n k k mV 2m k = n m mX−1 i=0 (−1)n(i+1) 2m i V n−m m−i V(m+n)(m−i) + 2m m 2 n−m !
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[1] Gould, H.W.: Combinatorial identities. Morgantown. W. Va. (1972).[2] Kılıç, E., Ömür, N. and Türker Uluta¸s, Y.: Alternating sums of the powers of Fibonacci and Lucas numbers. Miskolc Mathematical Notes. 12 (1), 87-103 (2011).
[3] Kılıç, E., Türker Uluta¸s, Y. and Ömür, N.: Formulas for weighted binomial sums with the powers of terms of binary recurrences. Miskolc Mathematical Notes. 13 (1), 53-65 (2012).
[4] Kılıç, E., Ömür, N. and Koparal, S.: On alternating weighted binomial sums with falling factorials. Bulletin of Mathematical Analysis and Applications. 9 (1), 58-64 (2017).
[5] Kılıç, E., Ömür, N. and Koparal, S.: Formulae for two weighted binomial identities with the falling factorials. Ars Combinatoria. 138, 223-231 (2018).
[6] Khan, M. and Kwong, H.: Some binomial identities associated with the generalized natural number sequence. The Fibonacci Quarterly. 49 (1), 57-65 (2011).
[7] Kılıç, E. and Ta¸sdemir, F.: On binomial double sums with Fibonacci and Lucas Numbers-I. Ars Combinatoria. 144, 173-185 (2019).
[8] Kılıç, E. and Ta¸sdemir, F.: On binomial double sums with Fibonacci and Lucas Numbers-II. Ars Combinatoria. 144, 345-354 (2019).
[9] Ozeki, K.: On Melham’s sum. Fibonacci Quarterly. 46/47 (2), 107-110 (2008/09).
[10] Prodinger, H.: On a sum of Melham and its variants. Fibonacci Quarterly. 46/47 (3), 207-215 (2008/09).
[11] Wiemann, M. and Cooper, C.: Divisibility of an F-L type convolution, in Applications of Fibonacci numbers. ser. Proceedings of the 10th International Research Conference on Fibonacci Numbers and their Applications held at Northern Arizona University, Dorrecht: Klywer Acad. Publ. 9, 267-287 (2004
- ISSN: 2147-6268
- Yayın Aralığı: Yılda 4 Sayı
- Başlangıç: 2013
- Yayıncı: -