Degenerate Mittag-Leffler Function via the Degenerate Pochhammer Symbol

We introduce a coefficient-degenerate analogue of the three-parameter Mittag-Leffler function by replacing the classical rising factorial $(\gamma)_n$ in the numerator with the degenerate Pochhammer symbol $(\gamma)_{\lambda,n}$ associated with the degenerate gamma function $\Gamma_\lambda$, while keeping the standard scaling $\Gamma(\alpha n+\beta)$ in the denominator. Assuming an explicit nonresonance (pole-avoidance) condition that guarantees finiteness of $(\gamma)_{\lambda,n}$ for all $n\ge0$, we prove that the defining power series converges for every $z\in\C$ and hence defines an entire function $E^{[\lambda]}_{\alpha,\beta;\gamma}(z)$. The deformation depends continuously on $\lambda$ and recovers the Prabhakar function as $\lambda\to0^{+}$. We represent $E^{[\lambda]}_{\alpha,\beta;\gamma}$ as a Fox–Wright function and derive coefficient bounds and sharp growth estimates, including the order of the resulting entire function. For the special case $\alpha=1$, we obtain a closed form in terms of a degenerate confluent hypergeometric function, which is convenient for symbolic manipulation and computation. Using these representations, we establish shift relations for Riemann–Liouville fractional integrals and derivatives that parallel the classical Prabhakar calculus. A Laplace transform pair is proved and used to construct a one-parameter, closed-form deformation of the Havriliak–Negami transfer function, yielding analytically tractable relaxation kernels for non-Debye linear response in complex media. Finally, we develop a numerically stable evaluation based on rigorous a posteriori truncation bounds and illustrate the impact of coefficient degeneracy through computational experiments.