Fibonacci-based Pascal-like matrices with double factorial binomial structure

Abstract In this paper, we introduce and investigate a new class of Pascal-like matrices constructed using double F-factorial binomial coefficients, in which the standard factorial function is replaced by the Fibonacci factorial. We first define the double F-factorial binomial coefficient and employ it to generate a bivariate polynomial sequence incorporating both Fibonacci and Lucas factorial terms. Fundamental recurrence relations for the coefficients of these polynomials are established. Using these relations, we construct a $F$-Pascal-like matrix, denoted by $P_{t}\left ( u,v\right )_{F},$ and derive several of its algebraic properties, including matrix factorization, power identities, and determinant-trace relationships. We further define a new functional matrix $Q_{t,\lambda }\left ( x,y\right )_{F},$ referred to as the $F$ -Pascal-like matrix, and include its additive and product properties. The proposed constructions extend classical binomial and Pascal matrices within the framework of Fibonacci analysis, providing potential applications in combinatorics, number theory, and matrix theory.