- Mathematical Sciences and Applications E-Notes
- Vol: 8 Issue: 2
- On Binomial Sums and Alternating Binomial Sums of Generelized Fibonacci Numbers with Falling Factori...
On Binomial Sums and Alternating Binomial Sums of Generelized Fibonacci Numbers with Falling Factorials
Authors : Sibel Koparal, Neşe Ömür
Pages : 123-129
Doi:10.36753/mathenot.708004
View : 6 | Download : 3
Publication Date : 2020-10-15
Article Type : Research
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