Degenerate Mittag-Leffler Functions Defined via the Degenerate Gamma Function and Applications to Fractional Maxwell-Zener Viscoelasticity

Time-dependent materials often show relaxation and creep over many decades in time. Fractional Maxwell and Zener models describe this behavior with a small number of parameters, and their response functions are written in terms of Mittag--Leffler kernels. In this paper we introduce a $\lambda$--deformed two-parameter Mittag--Leffler function by replacing the classical gamma denominator in the Mittag--Leffler series with the degenerate gamma function $\Gamma_{\lambda}$. Using a Beta-integral representation of $\Gamma_{\lambda}$, we give admissible parameters and determine the exact radius of convergence $R_{\lambda}(\alpha)=|\lambda^{\alpha}|^{-1}$, which yields a sharp disk of analyticity. We also prove that $E^{(\lambda)}_{\alpha,\beta}$ converges to the classical Mittag--Leffler function acts as a memory-shape control that can improve fits to relaxation/creep data, while the standard fractional models are recovered in the limit $\lambda\to0^{+}$. 2020 Mathematics Subject Classification. 33E12; 33B15; 34A08; 74D05; 44A10; 26A33.