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

Source abstract

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 s u , F , ψ ( x ) and explicitly demonstrate their degenerate sequences. Furthermore, we extend this construction to matrix theory by defining the degenerate Fibo–Bernoulli matrix , the generalized degenerate Fibo–Pascal matrix , and the degenerate Fibo–Euler matrix . 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.