Supersymmetric Fibonacci polynomials

Abstract

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.