Relations between Chebyshev, Fibonacci and Lucas polynomials via trigonometric sums

Abstract

In this paper we derive some new identities involving the Fibonacci and Lucas polynomials and the Chebyshev polynomials of the first and the second kind. Our starting point is a finite trigonometric sum which equals the resolvent kernel on the discrete circle with $m$ vertices and which can be evaluated in two different ways. An expression for this sum in terms of the Chebyshev polynomials was deduced in \cite{JKS} and the expression in terms of the Fibonacci and Lucas polynomials is deduced in this paper. As a consequence, we establish some further identities involving trigonometric sums and Fibonacci, Lucas, Pell and Pell-Lucas polynomials and numbers, thus providing a "physical" interpretation for those identities. Moreover, the finite trigonometric sum of the type considered in this paper can be related to the effective resistance between any two vertices of the $N$-cycle graph with four nearest neighbors $C_{N}(1,2)$. This yields further identities involving Fibonacci numbers.