Chebyshev Polynomials and Fibonacci Numbers

The Chebyshev polynomials arise in several mathematical contexts such as approximation theory, numerical integration, and differential equations. Here we study a combinatorial interpretation of Chebyshev polynomials due to Shapiro, and we use it to give a slight variation of a combinatorial proof of Binet\u27s Formula due to Benjamin, Derks and Quinn. Another beautiful formula for the Fibonacci numbers involves complex roots of unity. Presently, no combinatorial proof is known. We give combinatorial proofs of some related identities as progress toward a full combinatorial proof