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Binomial Fibonacci Sums from Chebyshev Polynomials
Kunle Adegoke
Department of Physics and Engineering Physics
Obafemi Awolowo University
220005 Ile-Ife
Nigeria
adegoke00@gmail.com
in Robert Frontczak
Reutlingen
Germany
robert.frontczak@web.de
in Taras Goy
Faculty of Mathematics and Computer Science
Vasyl Stefanyk Precarpathian National University
Shevchenka Str. 57
76018 Ivano-Frankivsk
Ukraine
taras.goy@pnu.edu.ua

in

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 $\frac{k}{n+k}\binom{n+k}{n-k}$. We derive the identities by relating the underlying sums to Chebyshev polynomials. Finally, we study some combinatorial sums and derive a connection with a recent paper by Chu and Guo from 2022.