Spectral properties of a functional binomial matrix

Year 2024, Volume: 73 Issue: 3, 749 - 764

Morteza Bayat

https://doi.org/10.31801/cfsuasmas.1360864

Abstract

In this article, we consider the definition of the Fibonacci polynomial sequence with the second-order linear recurrence relation, where coefficients and initial conditions depend on the variable $t$. And then, we introduce the functional binomial matrix depending on the coefficients of the second-order linear recurrence relation. In the following, we study the spectral properties of the functional binomial matrix using the Fibonacci polynomial sequence and we obtain a diagonal decomposition for it using the Vandermunde matrix. Finally, by applying some linear algebra tools we obtain a number of combinatorial identities involving the Fibonacci polynomial sequence.

Keywords

Functional Fibonacci matrix, generalized Fibonacci sequence, generalized Fibonacci polynomial, characteristic polynomial, Pascal matrix, functional binomial matrix

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