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

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),