Nth-Level Prabhakar Fractional Derivative

• The Nth-level Prabhakar fractional derivative is a generalized operator built on the three-parameter Mittag-Leffler function, unifying Riemann–Liouville, Caputo, and Hilfer derivatives via iterated convolution kernels.
• It employs a recursive framework with adjustable memory and interpolation parameters, enabling explicit Laplace-transform solutions for complex time-fractional systems.
• Its rich algebraic structure and semigroup properties underpin practical applications in anomalous diffusion, viscoelasticity modeling, and dielectric response analyses.

The Nth-level Prabhakar fractional derivative is a class of fractional differential operators that generalize a spectrum of fractional calculus models, including Riemann–Liouville, Caputo, and Hilfer types. These operators are intimately connected to the three-parameter Mittag-Leffler function and enable the modeling of complex memory effects in time-fractional systems. The formalism introduces iterated (Nth-level) convolutional operators governed by a hierarchy of memory and interpolation parameters, with a rich algebraic and operational structure suitable for analytical and applied mathematics, as well as physical modeling in areas such as anomalous diffusion and viscoelasticity (Waheed et al., 24 Dec 2025, Fernandez et al., 2018, Giusti, 2019).

1. The Three-Parameter Mittag-Leffler Function and Prabhakar Kernel

The Prabhakar calculus is built upon the three-parameter Mittag-Leffler function: $$E_{\alpha,\beta}^{\gamma}(z) = \sum_{k=0}^\infty \frac{(\gamma)_k}{k! \, \Gamma(\alpha k + \beta)} z^k$$ where $(\gamma)_k = \gamma (\gamma+1)\cdots(\gamma+k-1)$ is the Pochhammer symbol, $\alpha,\beta,\gamma \in \mathbb{C}$ with $\Re(\alpha) > 0$.

The associated Prabhakar kernel is: $$e_{\alpha,\beta}^{\gamma}(t) = t^{\beta-1}E_{\alpha,\beta}^{\gamma}(\omega t^{\alpha}), \qquad t > 0$$ with $\omega \in \mathbb{C}$ a parameter controlling the memory depth (Fernandez et al., 2018, Polito et al., 2015).

2. Definitions: Prabhakar Integral, Derivative, and Nth-Level Operators

Prabhakar fractional integral: For $f \in L^1(a, b)$ and $\Re(\alpha) > 0$, $\Re(\beta) > 0$,

$$I_{a+}^{\alpha,\beta,\gamma}f(x) = \int_a^x (x-t)^{\beta-1} E_{\alpha,\beta}^{\gamma}(\omega (x-t)^{\alpha}) f(t) \, dt$$

This operator interpolates between different integral kernels and encompasses the Riemann–Liouville case ($\gamma=0$).

Prabhakar (Riemann-Liouville type) derivative:

If $m = \lceil \Re(\beta)\rceil$,

$$D_{a+}^{\alpha,\beta,\gamma}f(x) = \frac{d^m}{dx^m} I_{a+}^{\alpha, m-\beta, -\gamma} f(x)$$

Nth-level Prabhakar derivative:

Given $n \in \mathbb{N}$, iterating the Prabhakar operator $n$ times with possibly varying fractional parameters $s_i$ and interpolation weights $\theta_i$, the left-sided Nth-level Prabhakar fractional derivative is defined recursively as (Waheed et al., 24 Dec 2025): $$\begin{aligned} {}_{a}D_{x}^{(s_1,\dots,s_n),\,\gamma,\delta;\,(\theta_1,\dots,\theta_n)} f(x) = {}_aE_{x}^{s_n, -\gamma(1-\theta_n), \delta} \frac{d}{dx} \cdots {}_aE_{x}^{s_{1}, -\gamma(1-\theta_{1}), \delta} \frac{d}{dx} f(x) \end{aligned}$$ This formalism recovers RL, Caputo, Hilfer, and one-level Prabhakar derivatives as special cases. The right-sided version is analogous but reverses the convolution/differentiation order. The associated Prabhakar fractional integral is (Waheed et al., 24 Dec 2025): $${}_aE_x^{\alpha,\beta,\delta} f(x) = \int_a^x (x-t)^{\alpha-1} E_{\beta,\alpha}^\delta (\gamma (x-t)^\beta) f(t) \, dt$$ with the three-parameter Mittag-Leffler function as kernel.

3. Algebraic Structure and Semigroup Properties

A critical structural feature is the additive semigroup property: $${}_aE_{x}^{\alpha,\beta_1,\delta} \circ {}_aE_{x}^{\alpha,\beta_2,\delta} = {}_aE_{x}^{\alpha,\beta_1+\beta_2,\delta}$$ This allows the nth-level operator to be written as a single Prabhakar operator with scaled parameters: $$(\mathcal{E}_{\alpha,\beta}^\gamma)^N = \mathcal{E}_{\alpha, N\beta}^{N\gamma}$$ The Nth-level Prabhakar derivative thereby takes the form: $$\bigl(D^\gamma_{\rho,\mu,\omega,a^+}\bigr)^N f = D^{N\gamma}_{\rho, N\mu, \omega, a^+} f$$ or, in integral notation (Polito et al., 2015): $$\frac{d^M}{dx^M} \int_a^x (x-t)^{M - N\mu - 1} E_{\rho, M-N\mu}^{-N\gamma}(\omega(x-t)^\rho) f(t) \, dt$$ where $M = \lceil N \Re(\mu)\rceil$. This semigroup composition mirrors the commutative structures found in classical fractional calculus and underlies the developability of operational calculus for these operators (Waheed et al., 24 Dec 2025, Polito et al., 2015, Giusti, 2019).

4. Analytical Properties and Special Cases

The Nth-level Prabhakar derivative is linear and its kernel is weakly singular but locally integrable. The operator’s Laplace transform is central for both theoretical analysis and applied solution schemes: $$\mathcal{L} \bigl\{ D_{a+}^{n;\alpha,\beta,\gamma} f \bigr\}(s) = s^{n\beta}\left(1-\omega s^{-\alpha}\right)^{n\gamma} \mathcal{L}\{f\}(s)$$ With appropriate initial-value terms handled as in the classical RL case (Fernandez et al., 2018, Giusti, 2019). Parameter constraints are $\Re(\alpha)>0$, $\Re(\beta)>0$, and for order/integrability, $\Re(N\beta)>0$. Series representations in terms of RL integrals are available: $$\mathcal{E}_{\alpha, N\beta}^{N\gamma} f = \sum_{k=0}^\infty \binom{N\gamma}{k} \omega^k I_{0+}^{\alpha k + N\beta} f$$ recovering RL/Caputo when $(\gamma, \omega)\to(1,0)$ (Fernandez et al., 2018, Polito et al., 2015).

Special cases include:

• $\gamma=0, \omega=0$: RL/Caputo operators.
• One-level $(N=1)$: Standard Prabhakar derivative.
• Integer order $N\mu$: Reduces to a classical $k$th derivative (for $k \in \mathbb{N}$) acting after a single Prabhakar-type convolution (Polito et al., 2015, Giusti et al., 2020, Waheed et al., 24 Dec 2025).

5. Operational Calculus and Solution of Differential Equations

A Mikusiński-type operational calculus framework (in the field of convolution quotients) is established for Nth-level Prabhakar operators (Waheed et al., 24 Dec 2025). In this setting, the operational inverse of the Prabhakar kernel allows algebraic manipulation of fractional differential equations, yielding explicit solution representations: $$y(x) = \sum_{m=0}^\infty A^{\, m} \left[ f * x^{\alpha m-1} E_{\gamma,\alpha m}^{m\delta}(\gamma\, x^\beta) \right](x) + \sum_{k=0}^{n-1} C_k x^{\alpha-k-1} E_{\gamma,\alpha-k}^{\delta} (\gamma x^\beta)$$ where $*$ denotes convolution, and $C_k$ are initial-condition-dependent constants. This approach bypasses direct inversion techniques or special-function expansions, providing tractable analytic solutions for a variety of fractional ODEs and time-fractional PDEs, as exemplified by the time-fractional heat equation (Waheed et al., 24 Dec 2025).

6. Applications and Hierarchy of Special Cases

The Nth-level Prabhakar derivative admits wide-ranging applications, including:

• Linear and nonlinear viscoelasticity modeling with history-dependent relaxation
• Anomalous diffusion and heat conduction with memory kernels
• Electrochemical and dielectric response theories with generalized fractional response functions (Garra et al., 2017, Fernandez et al., 2018, Waheed et al., 24 Dec 2025)

The operator’s structure permits:

• Recovery of RL, Caputo, Hilfer, and standard Prabhakar models via choices of $(n, \theta_i, \gamma, \delta)$
• Construction of product and chain rules at level $n$ directly from those at level 1 by scaling parameters
• Explicit Laplace-domain solution techniques, systematically incorporating both power-law and Prabhakar-type memory effects (Waheed et al., 24 Dec 2025, Fernandez et al., 2018)

7. Theoretical and Practical Implications

The Nth-level Prabhakar framework unifies a broad class of fractional derivatives and integrals, offering:

• Semigroup and operational structures compatible with classical theories
• Explicit convolution kernels, series, and Laplace transforms for analytic, computational, and numerical work
• Flexibility in function spaces (with appropriate integrability/smoothness for RL/Caputo variants)
• Solution formulas for linear fractional differential and integro-differential equations directly in operational form

A plausible implication is the substantial simplification in handling fractional evolution equations with nested or multi-scale memory, since the operational calculus allows direct algebraic manipulation and systematic handling of initial conditions. The approach readily extends to applications in renewal theory, control, and signal-processing models with distributed memory kernels (Waheed et al., 24 Dec 2025, Giusti et al., 2020, Fernandez et al., 2018).

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Selected References:

• "Operational Calculus for the nth-Level Prabhakar Type Fractional Derivative with Applications" (Waheed et al., 24 Dec 2025)
• "Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions" (Fernandez et al., 2018)
• "General fractional calculus and Prabhakar's theory" (Giusti, 2019)
• "Some Properties of Prabhakar-type Fractional Calculus Operators" (Polito et al., 2015)
• "A practical guide to Prabhakar fractional calculus" (Giusti et al., 2020)
• "The Prabhakar or three parameter Mittag--Leffler function: theory and application" (Garra et al., 2017)