New fractional integral unifying six existing fractional integrals

TL;DR

This paper introduces a new fractional integral that unifies six well-known fractional integrals into a single generalized form, providing a comprehensive framework with additional properties.

Contribution

The paper presents a novel fractional integral that generalizes six existing integrals, along with properties like semigroup, boundedness, shift, and integration-by-parts formulas.

Findings

Unified fractional integral form encompassing six integrals

Derived properties including semigroup and boundedness

Established formulas for shift and integration-by-parts

Abstract

In this paper we introduce a new fractional integral that generalizes six existing fractional integrals, namely, Riemann-Liouville, Hadamard, Erd\'elyi-Kober, Katugampola, Weyl and Liouville fractional integrals in to one form. Such a generalization takes the form \[ \left({}^{\rho}\mathcal{I}^{\alpha, \beta}_{a+;\eta, \kappa}f\right)(x)=\frac{\rho^{1-\beta}x^{\kappa}}{\Gamma(\alpha)}\int_a^x \frac{\tau^{\rho \eta +\rho-1}}{(x^\rho-\tau^\rho)^{1-\alpha}}f(\tau)\text{d}\tau, \quad 0\leq a < x < b \leq \infty. \] A similar generalization is not possible with the Erd\'elyi-Kober operator though there is a close resemblance with the operator in question. We also give semigroup, boundedness, shift and integration-by-parts formulas for completeness.