Determinants of Toeplitz–Hessenberg matrices with generalized Fibonacci entries Article Swipe

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Fibonacci number Toeplitz matrix Mathematics Fibonacci polynomials Simple (philosophy) Combinatorics Mathematical proof Lucas number Sign (mathematics) Combinatorial proof Chebyshev polynomials Recurrence relation Chebyshev filter Discrete mathematics Algebra over a field Pure mathematics Orthogonal polynomials Classical orthogonal polynomials Mathematical analysis Philosophy Geometry Epistemology

Taras Goy , Mark Shattuck ·

YOU? · · 2019 · Open Access · · DOI: https://doi.org/10.7546/nntdm.2019.25.4.83-95 · OA: W2996567816

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 S n are given for our formulas in several particular cases, including those involving the Chebyshev polynomials.