Binomial Fibonacci sums from Chebyshev polynomials

Abstract

We explore new types of binomial sums with Fibonacci and Lucas numbers. The binomial coefficients under consideration are $\frac{n}{n+k}\left(\genfrac{}{}{0ex}{}{n+k}{n-k}\right)$ and $\frac{k}{n+k}\left(\genfrac{}{}{0ex}{}{n+k}{n-k}\right)$. The identities are derived by relating the underlying sums to Chebyshev polynomials. Finally, some combinatorial sums are studied and a connection to a recent paper by Chu and Guo from 2022 is derived.

25 pages