On Binomial Sums and Alternating Binomial Sums of Generelized Fibonacci Numbers with Falling Factorials

Year 2020, , 123 - 129, 15.10.2020

Sibel Koparal , Neşe Ömür

https://doi.org/10.36753/mathenot.708004

Abstract

In this paper, we consider and obtain binomial sums and alternating binomial
sums including falling factorial of the summation indices. For example, for
nonnegative integer $m,$
\begin{eqnarray*}
&&\sum\limits_{k=0}^{n}\dbinom{n}{k}k^{\underline{m}}U_{2k}^{2m}=\frac{n^{\underline{m}}}{\left( p^{2}+4\right) ^{m}}\left(
\sum\limits_{i=0}^{m}\left( -1\right) ^{i}\dbinom{2m}{i}V_{2\left(
m-i\right) }^{n-m}V_{2\left( m+n\right) \left( m-i\right) }-\left( -1\right)
^{m}2^{n-m}\dbinom{2m}{m}\right),

Keywords

Generalized Fibonacci numbers, sums, falling factorials

References

• Reference1 Gould, H.W.: Combinatorial identities, Morgantown, W. Va., (1972).
• Reference2 Kılıç E., Ömür, N. and Türker Ulutaş, Y.: Alternating sums of the powers of Fibonacci and Lucas numbers. Miskolc Mathematical Notes. 12 (1), 87-103 (2011).
• Reference3 Kılıç E., Türker Ulutaş, Y., Ömür, N.: Formulas for weighted binomial sums with the powers of\ terms of binary recurrences. Miskolc Mathematical Notes. 13 (1), 53-65 (2012).
• Reference4 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).
• Reference5 Kılıç E., Ömür, N.. and Koparal, S.: Formulae for two weighted binomial identities with the falling factorials.} Ars Combinatoria. 138 223-231 (2018).
• Reference6 Khan M. and Kwong H.: Some binomial identities associated with the generalized natural number sequence. The Fibonacci Quarterly. 49 (1), 57-65 (2011).
• Reference7 Kılıç E., and Taşdemir, F.:On binomial double sums with Fibonacci and Lucas Numbers-I. Ars Combinatoria. 144, 173-185 (2019).
• Reference8 Kılıç E., and Taşdemir, F.: On binomial double sums with Fibonacci and Lucas Numbers-II. Ars Combinatoria. 144, 345-354 (2019).
• Reference9 Ozeki, K.: On Melham's sum. Fibonacci Quarterly. 46/47 (2), 107-110 (2008/09).
• Reference10 Prodinger, H.: On a sum of Melham and its variants. Fibonacci Quarterly. 46/47 (3), 207-215 (2008/09).
• Reference11 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).