  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.
