On the ψ− Sheffer F− Polynomials and Their Degenerate Matrices

In this study, we systematically construct a novel framework within umbral calculus by integrating the structures of generalized F-calculus and degenerate sequences. Utilizing Roman’s foundational approach to umbral calculus, we first introduce a new family of ψ−F− linear functionals and their corresponding ψ−F− differential operators on the polynomial space. Based on these operational tools, we define a generalized class of polynomials designated as the ψ− Sheffer F− polynomials su,F,ψx and explicitly demonstrate their degenerate sequences. Furthermore, we extend this construction to matrix theory by defining the degenerate Fibo–Bernoulli matrix Bx,F,ψ, the generalized degenerate Fibo–Pascal matrix Pu+1x,F,ψ, and the degenerate Fibo–Euler matrix EF,ψ. Finally, we establish the algebraic relationships and structural connections among these newly introduced degenerate matrices. The results obtained herein provide an organized algebraic bridge between degenerate Sheffer sequences and Fibonacci-based matrix representations.