Determinants of Toeplitz-Hessenberg matrices with generalized Fibonacci entries

Abstract

In this paper, we evaluate several families of Toeplitz–Hessenberg matrices whose entries are generalized Fibonacci numbers. In particular, we find simple formulas for several determinants whose entries are translates of the Chebyshev polynomials of the second kind. Equivalently, these determinant formulas may also be rewritten as identities involving sums of products of generalized Fibonacci numbers and multinomial coefficients. Combinatorial proofs which make use of sign-reversing involutions and the definition of a determinant as a signed sum over the symmetric group Sn are given for our formulas in several particular cases, including those involving the Chebyshev polynomials.