Degenerate-Pochhammer-Based Truncated M-Fractional Operators with Applications to RC Relaxation

Real dielectric media and non-ideal circuit elements often exhibit relaxation behaviors that deviate from the classical exponential response of an ideal RC network.Fractional-order models have therefore become a standard tool for capturing anomalous (non-exponential) relaxation and for improving calibration against measured data. In this work, we introduce a degenerate-Pochhammer-based family of truncated $M$-fractional operators built from the degenerate gamma function and degenerate Pochhammer symbols.After a normalization that annihilates constants, we prove that the associated derivative is local on $C^{1}$ and admits the explicit representation\[{}^{i}\!D^{\alpha}_{M,\lambda} f(t)=K_{\lambda}\,t^{1-\alpha} f'(t),\qquad 0<\alpha\le 1,\]where $K_{\lambda}$ is an explicit constant determined by the first-order coefficient of the normalized kernel.In particular, for the parameter regime $p=q=0$, $\beta=\gamma=1$, and $i=1$, our construction recovers the conformable fractional derivative exactly. Using the operational time $s(t)=t^{\alpha}/(\alpha K_{\lambda})$, initial value problems written with ${}^{i}\!D^{\alpha}_{M,\lambda}$ reduce to ordinary differential equations in $s$.As a canonical application, we revisit the first-order resistor--capacitor (RC) relaxation model and obtain closed-form responses; for a step input $u(t)\equiv V_{0}$ we derive$v(t)=V_{0}+(v_{0}-V_{0})\exp\!\bigl(-t^{\alpha}/(\alpha K_{\lambda}RC)\bigr)$ and interpret $K_{\lambda}$ as an effective time-constant scaling.We also illustrate the reduction on lumped thermal dynamics and on damped/forced oscillators, and we include numerical examples. MSC Classification: 26A33 , 34A08 , 33B15