This paper introduces and systematically investigates Fibonacci-based analogues and generalizations of degenerate Stirling and Lah numbers. We begin by recalling the classical definitions and key properties of Stirling numbers of both kinds, Lah numbers, Fibonacci numbers, and Fibonomial coefficients, along with F-falling and F-rising factorials. The foundational concept of degenerate numbers and their associated degenerate factorials, as initiated by Carlitz, is also reviewed. Our primary contribution is the definition of four new families of numbers: the degenerate F-Stirling numbers of the first kind and \(S_{1,\lambda }^F(n,k)\) , the degenerate F-Stirling numbers of the second kind \(S_{2,\lambda }^F(n,k)\) , and the degenerate F-Lah numbers \(L_{\lambda }^F(n,k)\) . These numbers are precisely characterized as connection coefficients between the standard power basis \(\{x^n: n \ge 0\}\) and the F-falling and F-rising factorial bases, and their degenerate counterparts. We derive fundamental recurrence relations for each class of these new numbers, providing a structural foundation for their analysis. Furthermore, we establish their generating functions and prove several combinatorial identities, including inverse relations. Our methodology primarily utilizes generating function techniques and properties of degenerate Fibonacci-based exponential and logarithmic functions. This work extends the rich theory of degenerate combinatorial numbers into the realm of Fibonacci sequences, offering new insights into their algebraic and combinatorial structures.
