  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.
