General Fractional Integrals (GFIs) Overview

• General Fractional Integrals (GFIs) are kernel-based nonlocal operators that extend classical integration beyond the Riemann–Liouville paradigm.
• GFIs employ versatile kernel constructions—including Sonine pairs, analytic-kernel families, and parametric deformations—to recover classical operators and reveal new analytic structures.
• They provide unified frameworks for numerical approximation, probabilistic interpretations, and inequality formulations in modern fractional calculus.

General fractional integrals (GFIs) are kernel-based nonlocal operators that extend classical fractional integration beyond the single power-law Riemann–Liouville paradigm. In current usage, the term does not denote one canonical operator but a family of related frameworks: convolution operators generated by Sonine or Luchko kernel pairs, parametric deformations interpolating between Riemann–Liouville and Hadamard calculus, hypergeometric- and analytic-kernel families, and operators defined with respect to another function $g$. Their common feature is an integral representation of the form

$(I f)(x)=\int_a^x K(x,u)\,f(u)\,du,$

with the kernel chosen so that semigroup laws, inverse relations with suitable derivatives, or transform-domain factorization remain available (Luchko, 2021, Gül et al., 2023).

1. Operator concept and kernel viewpoints

A central GFI model is the convolution operator

$(I(K)f)(t)=\int_0^t K(t-\tau)f(\tau)\,d\tau,$

which specializes to the Riemann–Liouville fractional integral when $K(t)=h_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$ (Luchko, 2021). A broader kernel formulation also appears in the form

$$(I_{0+}^{\rho,\theta}f)(x)=\int_0^x K(x,u;\rho,\theta)\,f(u)\,du,$$

where the parameter dependence is carried entirely by the kernel (Sibisi, 2023).

Another important class replaces the power kernel by a general analytic factor. For $A(z)=\sum_{n=0}^\infty a_n z^n$, analytic on a suitable disc, the left analytic-kernel GFI is

$$(AI_{a+}^{\alpha,\beta} f)(x)=\int_a^x f(t)\,(x-t)^{\alpha-1}A\big((x-t)^\beta\big)\,dt,$$

with a corresponding right-sided operator (Gül et al., 2023). In this setting the kernel expands as

$$(x-t)^{\alpha-1}A\big((x-t)^\beta\big) =\sum_{n=0}^\infty a_n (x-t)^{\alpha+\beta n-1},$$

so the operator becomes a series of Riemann–Liouville integrals of orders $\alpha+\beta n$ (Gül et al., 2023).

A further generalization transports fractional calculus through a monotone reference function $g$. The left-sided parametric GFI is

$(I f)(x)=\int_a^x K(x,u)\,f(u)\,du,$0

with an analogous right-sided operator (Tarasov, 2 Sep 2025). This realizes fractional integration in the variable $(I f)(x)=\int_a^x K(x,u)\,f(u)\,du,$1, not directly in $(I f)(x)=\int_a^x K(x,u)\,f(u)\,du,$2. The substitution operator $(I f)(x)=\int_a^x K(x,u)\,f(u)\,du,$3, defined by $(I f)(x)=\int_a^x K(x,u)\,f(u)\,du,$4, yields the exact conjugation formula

$(I f)(x)=\int_a^x K(x,u)\,f(u)\,du,$5

which makes the structural relation with finite-interval GFC explicit (Tarasov, 2 Sep 2025).

2. Sonine kernels, generalized Sonine pairs, and arbitrary order

The most systematic abstract framework in the cited literature is based on Sonine kernels. A pair $(I f)(x)=\int_a^x K(x,u)\,f(u)\,du,$6 is a Sonine pair if

$(I f)(x)=\int_a^x K(x,u)\,f(u)\,du,$7

with $(I f)(x)=\int_a^x K(x,u)\,f(u)\,du,$8 in the regularity class used by Luchko (Luchko, 2021). The associated GFI is

$(I f)(x)=\int_a^x K(x,u)\,f(u)\,du,$9

For arbitrary order, the Sonine condition is generalized to

$(I(K)f)(t)=\int_0^t K(t-\tau)f(\tau)\,d\tau,$0

and the admissible kernel pairs form the class $(I(K)f)(t)=\int_0^t K(t-\tau)f(\tau)\,d\tau,$1 (Luchko, 2021). In this setting, the GFI keeps the same convolution form,

$(I(K)f)(t)=\int_0^t K(t-\tau)f(\tau)\,d\tau,$2

while the associated general fractional derivatives (GFDs) are

$(I(K)f)(t)=\int_0^t K(t-\tau)f(\tau)\,d\tau,$3

and

$(I(K)f)(t)=\int_0^t K(t-\tau)f(\tau)\,d\tau,$4

(Luchko, 2021).

This formalism reproduces the classical Riemann–Liouville and Caputo operators by taking

$(I(K)f)(t)=\int_0^t K(t-\tau)f(\tau)\,d\tau,$5

It also admits non-power-law kernel pairs, including Bessel and modified Bessel kernels (Luchko, 2021).

Two general fundamental theorems are central. For $(I(K)f)(t)=\int_0^t K(t-\tau)f(\tau)\,d\tau,$6,

$(I(K)f)(t)=\int_0^t K(t-\tau)f(\tau)\,d\tau,$7

and, on the Caputo side,

$(I(K)f)(t)=\int_0^t K(t-\tau)f(\tau)\,d\tau,$8

(Luchko, 2021). These identities are the structural reason Sonine- and Luchko-based GFIs are widely treated as the kernel-theoretic core of general fractional calculus.

A notable point is the asymmetry between integral and derivative kernels. In Luchko’s arbitrary-order framework, the derivative kernel $(I(K)f)(t)=\int_0^t K(t-\tau)f(\tau)\,d\tau,$9 always has an integrable singularity at $K(t)=h_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$0, while the integral kernel $K(t)=h_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$1 may be singular or continuous at the origin, depending on the order (Luchko, 2021). This sharply distinguishes the general theory from a purely power-law intuition.

3. Major operator families and unification of classical calculus

Several concrete GFI families were introduced precisely to unify previously separate fractional integrals.

Before the table, it is useful to note that these families are not equivalent formulations of the same object. They are different parameterized operator classes, each chosen to recover specific classical cases or to expose specific analytic structures.

Family	Defining kernel/operator	Classical recoveries
Katugampola	$K(t)=h_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$2	RL as $K(t)=h_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$3, Hadamard as $K(t)=h_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$4
Five-parameter Katugampola-type	$K(t)=h_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$5	RL, Hadamard, Erdélyi–Kober, Katugampola, Weyl, Liouville
Saigo	$K(t)=h_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$6	RL when $K(t)=h_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$7, Erdélyi–Kober when $K(t)=h_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$8
Analytic-kernel	$K(t)=h_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$9	RL, Mittag–Leffler/Prabhakar, AB-type operators

The Katugampola operator was introduced in a form that interpolates between the additive Riemann–Liouville kernel and the logarithmic Hadamard kernel by varying a single deformation parameter $$(I_{0+}^{\rho,\theta}f)(x)=\int_0^x K(x,u;\rho,\theta)\,f(u)\,du,$$0 (Chen et al., 2016). A related earlier formulation used $$(I_{0+}^{\rho,\theta}f)(x)=\int_0^x K(x,u;\rho,\theta)\,f(u)\,du,$$1 instead of $$(I_{0+}^{\rho,\theta}f)(x)=\int_0^x K(x,u;\rho,\theta)\,f(u)\,du,$$2 and recovered Riemann–Liouville at $$(I_{0+}^{\rho,\theta}f)(x)=\int_0^x K(x,u;\rho,\theta)\,f(u)\,du,$$3 and Hadamard in the limit $$(I_{0+}^{\rho,\theta}f)(x)=\int_0^x K(x,u;\rho,\theta)\,f(u)\,du,$$4, with boundedness and semigroup results proved in weighted spaces $$(I_{0+}^{\rho,\theta}f)(x)=\int_0^x K(x,u;\rho,\theta)\,f(u)\,du,$$5 (Katugampola, 2010).

The five-parameter operator

$$(I_{0+}^{\rho,\theta}f)(x)=\int_0^x K(x,u;\rho,\theta)\,f(u)\,du,$$6

goes further: it unifies six named integrals—Riemann–Liouville, Hadamard, Erdélyi–Kober, Katugampola, Weyl, and Liouville—in a single kernel family, and it carries shift, semigroup, boundedness, and product-integration formulas (Katugampola, 2016). The Saigo family, with Gauss hypergeometric kernels, provides another classical axis of generalization, placing Riemann–Liouville and Erdélyi–Kober as parameter choices within a hypergeometric-kernel calculus (Rahman et al., 2016).

The analytic-kernel class is especially broad because the choice of $$(I_{0+}^{\rho,\theta}f)(x)=\int_0^x K(x,u;\rho,\theta)\,f(u)\,du,$$7 determines the model. The literature cited in the data explicitly states that Riemann–Liouville, Prabhakar- or Mittag–Leffler-type, AB-type, and related models arise as special cases for appropriate $$(I_{0+}^{\rho,\theta}f)(x)=\int_0^x K(x,u;\rho,\theta)\,f(u)\,du,$$8, $$(I_{0+}^{\rho,\theta}f)(x)=\int_0^x K(x,u;\rho,\theta)\,f(u)\,du,$$9, and $A(z)=\sum_{n=0}^\infty a_n z^n$0 (Gül et al., 2023).

4. Transform methods, probabilistic constructions, and special functions

Transform-domain factorization is one of the strongest organizing principles for GFIs. For the Katugampola family, Mellin transforms of the generalized fractional integrals and derivatives are available in closed form. For example, the left-sided Mellin transform satisfies

$A(z)=\sum_{n=0}^\infty a_n z^n$1

and the corresponding derivative formula has the reciprocal gamma-ratio structure (Katugampola, 2011). In the limits $A(z)=\sum_{n=0}^\infty a_n z^n$2 and $A(z)=\sum_{n=0}^\infty a_n z^n$3, these formulas reduce to the classical Mellin symbols of Riemann–Liouville and Hadamard operators (Katugampola, 2011).

A distinct line of development constructs GFI-type kernels probabilistically. Starting from the Riemann–Liouville integral of a one-sided $A(z)=\sum_{n=0}^\infty a_n z^n$4-stable density,

$A(z)=\sum_{n=0}^\infty a_n z^n$5

and then mixing over the scale parameter $A(z)=\sum_{n=0}^\infty a_n z^n$6 with a gamma distribution, one obtains

$A(z)=\sum_{n=0}^\infty a_n z^n$7

a weighted superposition of Riemann–Liouville integrals of stable densities at different scales (Sibisi, 2023). The central theorem identifies a specific four-parameter subfamily with the Prabhakar kernel,

$A(z)=\sum_{n=0}^\infty a_n z^n$8

thereby showing the Prabhakar function as a gamma-mixed family of Riemann–Liouville integrals (Sibisi, 2023). The same paper proves that the resulting Prabhakar function is the Laplace transform of a four-parameter distribution and recovers Pollard’s Mittag–Leffler law and the generalized two- and three-parameter Mittag–Leffler distributions as special cases (Sibisi, 2023).

GFIs also arise from fractional powers of Lie-group ladder operators. Fractional Weyl- and Riemann-type powers of operators such as $A(z)=\sum_{n=0}^\infty a_n z^n$9, $$(AI_{a+}^{\alpha,\beta} f)(x)=\int_a^x f(t)\,(x-t)^{\alpha-1}A\big((x-t)^\beta\big)\,dt,$$0, and $$(AI_{a+}^{\alpha,\beta} f)(x)=\int_a^x f(t)\,(x-t)^{\alpha-1}A\big((x-t)^\beta\big)\,dt,$$1 generate integral transforms that shift the order or degree of associated Legendre and Ferrers functions by non-integer amounts; the multi-derivative and multi-integral formulas for integer parameter shifts are obtained as the $$(AI_{a+}^{\alpha,\beta} f)(x)=\int_a^x f(t)\,(x-t)^{\alpha-1}A\big((x-t)^\beta\big)\,dt,$$2 special cases (Durand, 2021). In this setting, the GFI is not merely a kernel operator in one variable; it is a symmetry-generated transform acting on representation functions.

An alternative, series-based representation replaces the kernel integral by an infinite-order differential operator. By repeated integration by parts and analytic continuation, the paper on “Fractional integration and differentiation” derives

$$(AI_{a+}^{\alpha,\beta} f)(x)=\int_a^x f(t)\,(x-t)^{\alpha-1}A\big((x-t)^\beta\big)\,dt,$$3

and a companion series for $$(AI_{a+}^{\alpha,\beta} f)(x)=\int_a^x f(t)\,(x-t)^{\alpha-1}A\big((x-t)^\beta\big)\,dt,$$4 (Yaremko et al., 2023). This is a different generalization route: it preserves the order parameter $$(AI_{a+}^{\alpha,\beta} f)(x)=\int_a^x f(t)\,(x-t)^{\alpha-1}A\big((x-t)^\beta\big)\,dt,$$5 but expresses the operator through a derivative series rather than through a primitive kernel.

5. Inequalities, approximation theory, and numerical realization

A large part of the modern GFI literature develops inequalities once at the level of a general operator family and then recovers the classical models by specialization.

For Katugampola integrals, Hermite–Hadamard and Hermite–Hadamard–Fejér inequalities were proved in a form that reduces to the Riemann–Liouville case as $$(AI_{a+}^{\alpha,\beta} f)(x)=\int_a^x f(t)\,(x-t)^{\alpha-1}A\big((x-t)^\beta\big)\,dt,$$6 and to the Hadamard case as $$(AI_{a+}^{\alpha,\beta} f)(x)=\int_a^x f(t)\,(x-t)^{\alpha-1}A\big((x-t)^\beta\big)\,dt,$$7 (Chen et al., 2016). The same work gives derivative-based error bounds, including estimates with $$(AI_{a+}^{\alpha,\beta} f)(x)=\int_a^x f(t)\,(x-t)^{\alpha-1}A\big((x-t)^\beta\big)\,dt,$$8 and with convex $$(AI_{a+}^{\alpha,\beta} f)(x)=\int_a^x f(t)\,(x-t)^{\alpha-1}A\big((x-t)^\beta\big)\,dt,$$9, for the deviation between arithmetic averages and generalized fractional means (Chen et al., 2016).

For analytic-kernel GFIs, reverse Minkowski and related inequalities were derived directly for the whole class

$$(x-t)^{\alpha-1}A\big((x-t)^\beta\big) =\sum_{n=0}^\infty a_n (x-t)^{\alpha+\beta n-1},$$0

under positivity assumptions on the coefficients of $$(x-t)^{\alpha-1}A\big((x-t)^\beta\big) =\sum_{n=0}^\infty a_n (x-t)^{\alpha+\beta n-1},$$1 (Gül et al., 2023). The results then specialize automatically to the Riemann–Liouville, Mittag–Leffler/Prabhakar, and generalized proportional cases (Gül et al., 2023). This removes the need to prove Minkowski-type statements operator by operator.

Within the Hadamard subclass, generalized Hadamard, Ostrowski, and Simpson inequalities for $$(x-t)^{\alpha-1}A\big((x-t)^\beta\big) =\sum_{n=0}^\infty a_n (x-t)^{\alpha+\beta n-1},$$2-Lipschitzian functions were established via the left and right Hadamard fractional integrals

$$(x-t)^{\alpha-1}A\big((x-t)^\beta\big) =\sum_{n=0}^\infty a_n (x-t)^{\alpha+\beta n-1},$$3

with the classical logarithmic-mean inequalities recovered at $$(x-t)^{\alpha-1}A\big((x-t)^\beta\big) =\sum_{n=0}^\infty a_n (x-t)^{\alpha+\beta n-1},$$4 (İscan, 2013).

On the numerical side, the weakly singular Riemann–Liouville kernel can be mapped to a Jacobi weight. After the affine transformation $$(x-t)^{\alpha-1}A\big((x-t)^\beta\big) =\sum_{n=0}^\infty a_n (x-t)^{\alpha+\beta n-1},$$5, the left Riemann–Liouville integral becomes

$$(x-t)^{\alpha-1}A\big((x-t)^\beta\big) =\sum_{n=0}^\infty a_n (x-t)^{\alpha+\beta n-1},$$6

which is then approximated by a fractional Gauss–Jacobi quadrature rule with weight $$(x-t)^{\alpha-1}A\big((x-t)^\beta\big) =\sum_{n=0}^\infty a_n (x-t)^{\alpha+\beta n-1},$$7 (Jahanshahi et al., 2017). The paper combines this with the Hale–Townsend algorithm for nodes and weights and with Diethelm’s predictor–corrector method, and applies the resulting scheme to fractional boundary-value and variational problems (Jahanshahi et al., 2017).

6. Dynamics, parametric extensions, and conceptual distinctions

GFIs have been used as the basic operators of general fractional dynamics. In that framework, nonlocality is encoded by Sonin or Luchko kernels, and equations with GFI or GFD terms plus periodic kicks are solved exactly at discrete times to obtain “general nonlocal maps” without approximation (Tarasov, 22 Sep 2025). For instance, a Caputo-type kicked equation yields a discrete convolution map whose memory weights are determined directly by the GFI kernel $$(x-t)^{\alpha-1}A\big((x-t)^\beta\big) =\sum_{n=0}^\infty a_n (x-t)^{\alpha+\beta n-1},$$8 (Tarasov, 22 Sep 2025). The same paper extends this construction to arbitrary order using Luchko’s generalized Sonine condition $$(x-t)^{\alpha-1}A\big((x-t)^\beta\big) =\sum_{n=0}^\infty a_n (x-t)^{\alpha+\beta n-1},$$9 (Tarasov, 22 Sep 2025).

The parametric extension with respect to another function $\alpha+\beta n$0 provides an additional layer of generality. Since

$\alpha+\beta n$1

the semigroup law, the RL–Caputo relation, and both fundamental theorems of fractional calculus are inherited from the finite-interval GFC by conjugation (Tarasov, 2 Sep 2025). Classical Riemann–Liouville operators are recovered with $\alpha+\beta n$2, Hadamard operators with logarithmic $\alpha+\beta n$3, and Erdélyi–Kober operators through power-law choices of $\alpha+\beta n$4 together with simple multiplicative weights (Tarasov, 2 Sep 2025).

A recurrent source of confusion is terminological. Not every operator called a “generalized fractional integral” in the literature is nonlocal in the usual kernel-memory sense. One paper introduces

$\alpha+\beta n$5

paired with a local conformable-type derivative

$\alpha+\beta n$6

and explicitly notes that this framework does not recover the classical Riemann–Liouville, Caputo, Hadamard, or Erdélyi–Kober operators (Akkurt et al., 2017). This illustrates a broader fact: “GFI” is now an umbrella term covering both genuinely nonlocal kernel operators and more local weighted-integral constructions.

Taken together, these developments show that the modern theory of general fractional integrals is best understood as a kernel calculus rather than as a single formula. Its core problems are the classification of admissible kernels, the preservation of inverse and semigroup structures, the identification of classical operators as special cases, and the transfer of these structures into probability, special functions, inequalities, numerics, and nonlocal dynamics (Luchko, 2021, Sibisi, 2023, Tarasov, 22 Sep 2025).