Binomial Fibonacci sums from Chebyshev polynomials

TL;DR

This paper investigates novel binomial sums involving Fibonacci and Lucas numbers, establishing identities through Chebyshev polynomials and connecting to recent combinatorial research.

Contribution

It introduces new binomial sum identities with Fibonacci and Lucas numbers derived via Chebyshev polynomials, expanding the mathematical understanding of these sequences.

Findings

Derived identities linking binomial sums to Chebyshev polynomials

Established connections between Fibonacci sums and recent combinatorial results

Explored combinatorial sums related to Fibonacci and Lucas numbers

Abstract

We explore new types of binomial sums with Fibonacci and Lucas numbers. The binomial coefficients under consideration are $\frac{n}{n+k}\binom{n+k}{n-k}$ and $\frac{k}{n+k}\binom{n+k}{n-k}$. 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.