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
\[\frac{k^m}{n} \binom{2n}{n-k}\]
for positive integers \(m\). We then compute weighted half binomial sums involving powers of generalized Fibonacci and Lucas numbers of the form
\[\sum\limits_{k=0}^{n} \binom{2n}{n+k} \frac{k^m}{n}X_{tk}^r,\]
where \(X_n\) either generalized Fibonacci or Lucas numbers, and \(t\) and \(r\) are integers, focusing on cases where \(1 \leq m \leq 6\). Furthermore, we outline a general methodology for computing these sums for larger values of \(m\).