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 \(E_{\alpha ,\beta }\) as \(\lambda \rightarrow 0^{+}\) . A Fox–Wright representation is derived, leading to hypergeometric reductions when \(\alpha =1\) and when \(\alpha \in \mathbb {N}\) . As an application, we formulate generalized fractional Maxwell and Zener viscoelastic laws in which the relaxation modulus and creep compliance are expressed through \(E^{(\lambda )}_{\alpha ,\beta }\) . The extra parameter \(\lambda \) acts as a memory-shape control that can improve fits to relaxation/creep data, while the standard fractional models are recovered in the limit \(\lambda \rightarrow 0^{+}\) .
