New Sums Identities in Weighted Catalan Triangle With the Powers of Generalized Fibonacci And Lucas Numbers

Abstract

In this paper, we investigate a generalized Catalan triangle defined by
$$\frac{k^m}{n}(\genfrac{}{}{0ex}{}{2n}{n-k})$$
for positive integers $m$. We then compute weighted half binomial sums involving powers of generalized Fibonacci and Lucas numbers of the form
$$\sum _{k=0}^n(\genfrac{}{}{0ex}{}{2n}{n+k})\frac{k^m}{n}{X}_{tk}^r,$$
where $X_n$ either generalized Fibonacci or Lucas numbers, and $t$ and $r$ are integers, focusing on cases where $1\le m\le 6$. Furthermore, we outline a general methodology for computing these sums for larger values of $m$.