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arxiv: 2605.22854 · v1 · pith:KUXQAYSTnew · submitted 2026-05-18 · 🧮 math.CV

Prabhakar function and unified fractional kinetic equation in bicomplex space

Urvashi Purohit Sharma , Ritu Agarwal This is my paper

Pith reviewed 2026-05-25 00:09 UTC · model grok-4.3

classification 🧮 math.CV
keywords bicomplex Prabhakar functionfractional kinetic equationanalyticityLaplace transformMellin transformbicomplex spaceMittag-Leffler functionfractional calculus
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The pith

The bicomplex Prabhakar function is analytic in a region of convergence and solves a unified fractional kinetic equation in bicomplex space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper defines a bicomplex extension of the Prabhakar function, a generalization of the Mittag-Leffler function used in fractional calculus. It establishes analyticity inside a nonempty region of convergence and derives integral representations, recurrence formulas, differential relations, and bicomplex Laplace and Mellin transforms. The work then shows that this function appears both in a fractional kinetic equation and in its solution within bicomplex space. A sympathetic reader would care because the construction supplies explicit solution forms for fractional dynamics when variables take values in the bicomplex numbers.

Core claim

We introduce the bicomplex Prabhakar function and prove it is analytic inside a nonempty region of convergence. We derive its integral representations, recurrence formulas, differential relations, and the bicomplex Laplace and Mellin transforms. We then analyze a fractional kinetic equation in which the bicomplex Prabhakar function appears both in the equation and in its solution.

What carries the argument

The bicomplex Prabhakar function, defined via a three-parameter series in bicomplex variables, which supplies both the form of the kinetic equation and its explicit solution.

Load-bearing premise

The bicomplex Prabhakar function remains analytic inside a nonempty region of convergence, permitting derivation of its integral representations, recurrence formulas, differential relations, and transforms.

What would settle it

An explicit series expansion or direct substitution showing that the proposed solution fails to satisfy the fractional kinetic equation inside the claimed region of convergence.

read the original abstract

The Mittag-Leffler type functions arise naturally in the solution of fractional order integral and differential equations, especially in the investigations of the fractional generalization of the kinetic equation. This article introduces a bicomplex extension of the Prabhakar function, a generalization of the Mittag-Leffler function commonly used in fractional calculus. We explore the analyticity and determine the region of convergence for this new bicomplex Prabhakar function. Several fundamental properties are established, including its integral representations, recurrence formulas, and differential relations. Furthermore, we compute the bicomplex Laplace and Mellin transforms of the function, which are useful for solving differential and integral equations. Finally, we analyze a fractional kinetic equation where the bicomplex Prabhakar function appears both in the equation and in its solution, demonstrating its applicability in complex systems involving fractional dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The manuscript introduces a bicomplex extension of the Prabhakar function (a generalization of the Mittag-Leffler function), determines its analyticity and region of convergence, establishes integral representations, recurrence formulas, differential relations, and computes its bicomplex Laplace and Mellin transforms. It then analyzes a fractional kinetic equation in which the bicomplex Prabhakar function appears both in the defining equation and in the explicit solution.

Significance. If the analyticity claim and the listed derivations hold in the bicomplex setting, the work supplies a new special function and an associated solution technique for fractional-order equations over a four-dimensional commutative algebra. This could be relevant to modeling in complex fractional dynamics, but the actual significance is difficult to gauge because the provided material supplies no explicit definitions, series expansions, or proofs against which the claims can be checked.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful summary of our manuscript. The primary concern appears to be the absence of explicit definitions, series expansions, and proofs in the material reviewed. We address this point directly below.

read point-by-point responses

  1. Referee: the provided material supplies no explicit definitions, series expansions, or proofs against which the claims can be checked.

    Authors: The full manuscript contains the series definition of the bicomplex Prabhakar function, the proof of its analyticity together with the explicit region of convergence, the integral representations, recurrence relations, differential properties, the derivations of the bicomplex Laplace and Mellin transforms, and the complete analysis of the unified fractional kinetic equation (including both the equation and its explicit solution in terms of the new function). All steps are accompanied by proofs. We believe the referee may have been provided only the abstract or a condensed excerpt; the complete arXiv preprint supplies the requested material in full. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines the bicomplex Prabhakar function explicitly and derives its region of convergence, integral representations, recurrence formulas, differential relations, and Laplace/Mellin transforms directly from that definition and the assumption of analyticity in a nonempty domain. The fractional kinetic equation application places the function in both the equation and solution as an explicit demonstration of utility rather than a fitted prediction or self-referential construction. No load-bearing steps reduce by construction to inputs, no self-citation chains are invoked to justify uniqueness or ansatzes, and the derivation remains self-contained without renaming known results or smuggling assumptions via prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated. The work consists of a definitional extension and claimed derivations.

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Reference graph

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