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Title: Fibonacci–Lucas identities and the generalized Trudi formula
Authors: Гой, Тарас Петрович
Shattuck, Mark
Keywords: Hessenberg matrix
Fibonacci numbers
Trudi formula
Lucas numbers
Issue Date: 2020
Citation: Goy, T. Fibonacci–Lucas identities and the generalized Trudi formula / T. Goy, M. Shattuck // Notes on Number Theory and Discrete Mathematics. – 2020. – 26 (3). – P. 203-217.
Abstract: In this paper, we evaluate determinants of several families of Hessenberg matrices having Fibonacci numbers as their nonzero entries. By the generalized Trudi formula, these determinant identities may be written equivalently as formulas for the Lucas numbers in terms of the Fibonacci. We provide both algebraic and combinatorial proofs of our determinant results. The former makes use of expansion along columns and induction, while the latter draws upon the definition of the determinant as a signed sum over the symmetric group and uses parity-changing involutions.
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