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.
Goy, Taras and Shattuck, Mark
Fibonacci and Lucas Identities from Toeplitz–Hessenberg Matrices,
Applications and Applied Mathematics: An International Journal (AAM), Vol. 14,
2, Article 5.
Available at: https://digitalcommons.pvamu.edu/aam/vol14/iss2/5