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Title: Fibonacci and Lucas identities using Toeplitz–Hessenberg matrices
Authors: Гой, Тарас Петрович
Keywords: Fibonacci sequence
Lucas sequence
Toeplitz-Hessenberg matrix
Trudi’s formula
multinomial coefficient
Issue Date: 2019
Citation: Goy, T. Fibonacci and Lucas identities using Toeplitz-Hessenberg matrices / T. Goy, M. Shattuck // Applications and Applied Mathematics. – 2019. – 14(2). – P. 699-715
Abstract: In this paper, we consider determinants for some families of Toeplitz–Hessenberg matrices having various translates of the Fibonacci and Lucas numbers for the nonzero entries. These determinant formulas may also be rewritten as identities involving sums of products of Fibonacci and Lucas numbers and multinomial coefficients. Combinatorial proofs are provided of several of the determinants which make use of sign-changing involutions and the definition of the determinant as a signed sum over the symmetric group. This leads to a common generalization of the Fibonacci and Lucas determinant formulas in terms of the so-called Gibonacci numbers.
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