Properties of Tangent-Fibonacci Polynomials via Golden F-Calculus

Abstract

In this paper, we introduce and systematically study a new class of two-variable Tangent-Fibonacci polynomials together with their corresponding numbers within the framework of Golden F-Calculus. By employing suitable generating functions, we establish several fundamental properties of these polynomials, including summation formulas, recurrence relations, symmetry identities, and F-derivative representations. We further explore their structural connections with the Stirling-Fibonacci numbers of the second kind and derive various convolution-type identities and explicit summation expressions. In addition, we propose new parametric extensions characterized by trigonometric-type generating functions, and investigate their analytical behavior using the F-differential operator and functional equation methods. Moreover, we obtain an explicit matrix representation that clarifies the relationship between the associated polynomial matrix and a generalized Pascal matrix via Fibonomial coefficients of the first kind. The developed framework not only extends the theory of Fibonacci-based special polynomials in a coherent and unified manner, but also provides a foundation for further applications in combinatorics, number theory, approximation theory, and matrix analysis.