Exploring Extended Degenerate Bell-Based Appell Polynomials via Fractional Operators: Properties, Determinant Representations, and Numerical Illustrations

This paper introduces a new family of extended degenerate Bell-based Appell polynomials by applying Euler’s integral as a fractional operator to the Appell-type degenerate Bell polynomials. Beginning with the exponential operational rule and employing the fractional operator framework, this paper derives the operational connection, generating function, explicit summation formula, determinant representation, complete four-step proof via Cramer’s rule, recurrence relations, and the monomiality principle with raising and lowering operators and rigorously verifies the commutation relation. Applications to the extended degenerate Bell–Bernoulli, Bell–Euler, and Bell–Genocchi polynomials are presented as three numbered examples, each with generating functions; operational connections; determinant representations; numerical first values; and graphical illustrations including surface plots, zero diagrams, and stacked zero figures.