Source-linked AI summary
Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions
Arran Fernandez, Dumitru Baleanu, H. M. Srivastava
TL;DR
Different fractional-calculus models use different operators, and some newer models are less general than Prabhakar but easier to compute. The paper represents Prabhakar and generalized Prabhakar operators as series of Riemann–Liouville integrals, then applies the results to product and chain rules and fractional iteration.
Problem
Fractional-calculus models use different operators, while some newer alternatives are less general than Prabhakar but easier to compute.
Source: §1 · p. 3Method
The paper derives series representations of Prabhakar and generalized Prabhakar operators in terms of classical Riemann–Liouville fractional integrals.
Source: §1 · p. 3, §6 · p. 11Results
The series representation supports product- and chain-rule analogues in the Prabhakar model and results for other fractional-calculus models.
Source: §1 · p. 3, §6 · p. 11Takeaways & Limitations
The Prabhakar model’s semigroup property in some parameters can be linked to fractional iteration.
Source: §6 · p. 11The series result adds no new result for the iterated AB differintegral, whose defining series is recovered by substitution.
Source: §2 · p. 5
Abstract
from arXiv · showhide
We consider an integral transform introduced by Prabhakar, involving generalised multi-parameter Mittag-Leffler functions, which can be used to introduce and investigate several different models of fractional calculus. We derive a new series expression for this transform, in terms of classical Riemann-Liouville fractional integrals, and use it to obtain or verify series formulae in various specific cases corresponding to different fractional-calculus models. We demonstrate the power of our result by applying the series formula to derive analogues of the product and chain rules in more general fractional contexts. We also discuss how the Prabhakar model can be used to explore the idea of fractional iteration in connection with semigroup properties.
1 Introduction
Fractional calculus generalizes differentiation and integration to non-integer orders, with multiple models using different operators for different purposes. The paper develops a Prabhakar-based series approach and connects it to other fractional models, product and chain rules, and fractional iteration.
1 Introduction
- Fractional calculus generalizes standard calculus to non-integer orders, and its different models use different formulas suited to different purposes.The Riemann–Liouville model is described as the classical, most commonly used model; the Caputo model instead places differentiation inside the integral and gives zero for the derivative of a constant.Source: §1 · pp. 1–2
- The Prabhakar operator, introduced for a singular integral equation, has been adapted as a fractional differintegral and has applications in diffusion, relaxation, and stochastic processes.The paper focuses on the Riemann–Liouville-type Prabhakar derivative, while noting that a Caputo-type version can also be defined.Source: §1 · pp. 2–3
- The Atangana–Baleanu and iterated AB models are presented as related models, with the latter interpreting an AB fractional integral through iteration.The introduction also notes that other models can be less general than Prabhakar while being easier to compute and more directly applicable.Source: §1 · p. 3
- The paper derives series formulae for Prabhakar and generalized Prabhakar differintegrals using only classical Riemann–Liouville fractional integrals.The authors also use the formula to verify known results, derive product- and chain-rule analogues, and expect it to simplify numerical computation.Source: §1 · p. 3
- The authors use the Prabhakar series results to obtain analogous results for other models and investigate iteration of the Prabhakar differintegral.They identify the CF, AB, and iterated AB models as special cases of Prabhakar.Source: §1 · p. 3
2 Series expressions
Theorem 2.1 expresses the generalised Prabhakar operator as a series of Riemann–Liouville (RL) fractional integrals. Special cases yield formulas for Prabhakar and AB differintegrals, while the series also supports results on operator relationships and semigroup properties.
Generalised Prabhakar series
- Under conditions (18), the generalised Prabhakar operator has a series representation in classical RL fractional integrals for L1 functions.The series on the right-hand side is locally uniformly convergent.Source: §2 · p. 4
Specific models
- Prabhakar integral and derivative formulas follow as special cases, with locally uniformly convergent series on the stated parameter and function domains.The AB derivatives of Riemann–Liouville and Caputo type also have series formulas, and their series are locally uniformly convergent.Source: §2 · p. 5
- Using the series definitions, Prabhakar integrals and derivatives share one differintegral formula, with switching between them requiring sign changes in ρ and β.The AB formulas were already proved in; here they demonstrate the generality of the result.Source: §2 · p. 5
Scope
- The series theorem adds no new result for the iterated AB differintegral, whose defining series is recovered by substitution.The text distinguishes this from the possibility of fractional-power iteration in the Prabhakar model.Source: §2 · p. 5
Consequences
- The series formula makes proofs of fundamental Prabhakar operator identities and relationships with classical fractional-calculus operators shorter than their original Fubini-based proofs.The text identifies results previously proved in,, and.Source: §2 · p. 5
- The Prabhakar operator has a semigroup property under the stated positive-real-part assumptions, and its integral has a left inverse independent of γ.The left inverse is equivalent to the Prabhakar fractional derivative under the stated assumptions.Source: §2 · pp. 6–7
3 Fractional extensions of the product rule
The generalised Prabhakar series formula is used to derive a product-rule analogue for fractional differintegrals. The result applies under stated regularity conditions and yields corresponding rules for more specific models, including AB.
Generalised Prabhakar product rule
- For functions of the form x^ζξ(x), with Re(ζ) > −1 and ξ holomorphic, the generalised Prabhakar operator satisfies a product-rule analogue under conditions (18).The condition applies to f, g, and fg.Source: §3 · p. 7
Derivation
- The derivation substitutes a Riemann–Liouville product-rule expression into the generalised Prabhakar series, with local uniform convergence needed to make the proof rigorous.The finite truncation of the Riemann–Liouville rule is used as a starting point.Source: §3 · pp. 7–8
Example
- With f(x) = e^ax, g(x) = x, and c = i∞, the example reduces the outer series to two non-trivial terms, m = 0 and m = 1.The example uses the known RL fractional integral of an exponential function.Source: §3 · p. 8
Specific models
- The generalised Prabhakar product rule yields product-rule corollaries for more specific models, including the AB model.The text identifies the AB result as equation (31).Source: §3 · p. 8
4 Fractional extensions of the chain rule
The paper derives a generalised Prabhakar chain rule from its fractional product rule and the classical Faà di Bruno formula. The result applies under stated smoothness and holomorphy conditions and can be specialized to other models under the Prabhakar umbrella.
4 Fractional extensions of the chain rule
- The new series formula derives the generalised Prabhakar chain rule, extending a related result previously available for the AB model.The derivation applies the product rule to f(g(x)) and the unit function, then uses the classical Faà di Bruno formula.Source: §4 · pp. 9–10
- The generalised Prabhakar operator satisfies a chain rule when g is smooth and f(g(x)) has the specified holomorphic form, under parameter conditions (18).The final sum ranges over the set defined in (37).Source: §4 · p. 9
- Theorem 4.1 and Proposition 2.3 yield chain-rule analogues for other specific fractional models, including the AB result.The paper identifies the Gaussian e^(−x^2) as an application of Theorem 4.1.Source: §4 · p. 10
5 Iteration of fractional models
The paper extends the idea of fractional iteration to Prabhakar operators and gives a unified series expression for the resulting differintegral. It also establishes a semigroup property in the iteration parameter.
5 Iteration of fractional models
- Fractional iteration extends the attempt to define a non-integer iterate of an operator beyond its natural-number iterations to operators already interpreted as fractional differintegrals.The paper explores this idea for operators in the Prabhakar model.Source: §5 · p. 10
- The νth iteration of the Prabhakar operator is defined for any complex ν under Re(α) > 0 and Re(β) > 0.For Re(νβ) > 0 it agrees with the iterated-integral formula, and at ν = −1 it agrees with the Prabhakar fractional derivative.Source: §5 · pp. 10–11
- The unified series expression for the iterated Prabhakar differintegral applies regardless of the sign of Re(νβ) and is locally uniformly convergent.The series formulation merges the definitions of fractional derivatives and integrals into one expression.Source: §5 · p. 11
- The iterated Prabhakar differintegral satisfies a semigroup property in ν for Re(α) > 0 and Re(β) > 0 on any L1 function.The stated identity is for parameters α, β, ω, and ρ in C.Source: §5 · p. 11
6 Conclusions
The paper reduces Prabhakar fractional models and their generalisations to series involving only Riemann–Liouville integrals. It presents this representation as a route to results across several fractional-calculus models, including product and chain rules and semigroup properties.
6 Conclusions
- Prabhakar fractional models and their generalisations can be reduced to series involving only Riemann–Liouville integrals.The paper attributes this result to locally uniformly convergent series representations of Mittag-Leffler functions and their generalisations.Source: §6 · p. 11
- The series results enable or streamline proofs of semigroup and inverse properties, as well as product and chain rules, for Prabhakar and other fractional-calculus models.The paper characterizes the product and chain rules as new results and the semigroup and inverse properties as previously known facts.Source: §6 · p. 11
- The Prabhakar model's semigroup property in some parameters is linked to the idea of fractional iteration.The paper identifies fractional iteration as an important concept to consider in fractional calculus.Source: §6 · p. 11