Supersymmetric Fibonacci polynomials

It has long been recognized that Fibonacci-type recurrence relations can be used to define a set of versatile polynomials {p(n)(z)} that have Fibonacci numbers and Chebyshev polynomials as special cases. We show that a tridiagonal matrix, which can be factored into the product AB of two special matrices A and B, is associated with these polynomials. We apply tools that have been developed to study the supersymmetry of Hamiltonians that have a tridiagonal matrix representation in a basis to derive a set of partner polynomials {p(n)((+))(z)} associated with the matrix product BA. We find that special cases of these polynomials share similar properties with the Fibonacci numbers and Chebyshev polynomials. As a result, we find two new sum rules that involve the Fibonacci numbers and their product with Chebyshev polynomials.