Mikusinski-Type Operational Calculus

• Mikusinski-type operational calculus is an algebraic framework based on convolution quotients that transforms differential, fractional, and functional equations into solvable algebraic relations.
• It unifies classical Laplace transform techniques with generalized methods including Prabhakar-type fractional derivatives, q-difference, and noncommutative extensions using symbolic operational rules.
• The framework enables direct symbolic solutions of boundary value problems via explicit inversion, convolution series, and partial-fraction decomposition.

Mikusinski-type operational calculus is a class of algebraic frameworks for operator-theoretic solution of differential, fractional, and functional equations, based fundamentally on convolution quotients and their fields. This approach, developed initially by J. Mikusinski, provides an algebraic alternative to the classical Laplace transform and enables a purely algebraic symbolic calculus encompassing differentiation, integration, convolutions, and associated transforms. The theory generalizes naturally to accommodate fractional, difference, and nonlocal operators, with extensive recent advances in its algebraic, analytic, and operational aspects. The framework is central to the unification of classical and modern operational calculi, notably including general and Prabhakar-type fractional derivatives, noncommutative extensions, and difference-differential structures.

1. Classical Structure: Convolution Algebra and Mikusinski Field

Let $C$ denote the $\C$-vector space of complex-valued continuous functions on $[0,\infty)$. The convolution product

$(f * g)(t) = \int_0^t f(s)\,g(t-s)\,ds$

endows $C$ with a commutative ring structure (without unit). Titchmarsh’s theorem ensures $(C,+,*)$ has no zero divisors, allowing the construction of the field of convolution quotients:

$$Q(C) = \left\{ \frac{f}{g} \ \middle| \ f,g\in C,\,g \ne 0 \right\}/\sim,\qquad (f_1,g_1)\sim(f_2,g_2) \iff f_1 * g_2 = f_2 * g_1.$$

Elements of $Q(C)$, called Mikusinski operators, encode operational transformations. The function $l(t)\equiv1$ (the "integration operator") admits an inverse $s=1/l$ ("differential operator"). Differentiation and integration are implemented algebraically:

$$s \cdot f = f',\quad s^{-1} \cdot f = \int_0^t f(\tau)\,d\tau.$$

The basic operational rules, including shift and convolution theorems, emerge naturally in $Q(C)$. For example, convolution with $e^{\alpha t}$ corresponds to $(s-\alpha)^{-1}$, and convolution inverses correspond to operational inverses of multipliers (Nishioka, 16 Jun 2025, Lomadze, 2016, Bengochea et al., 2013).

2. Algebraic Extensions: General Fractional and Prabhakar Operators

Mikusinski-type construction applies to fractional calculus by considering weighted convolution kernels:

• For the Riemann–Liouville and Caputo kernels, one defines $h_\alpha(t) = t^{\alpha-1}/\Gamma(\alpha)$ and uses the quotient/convolution field $M_{-1}$, the field of convolution quotients over spaces such as $C_{-1}$ (functions with mild singularity at $t=0$) (Ahmad et al., 2023, Luchko, 2021).

The general framework for fractional derivatives with Sonine kernels $(K, k)$ identifies operators $I_K$, $D_k$ as convolution with $K$ (GFI) and $k$ (GFD) respectively, with the Caputo variant ${}^C D_k f = (k * f)(t) - f(0)\,k(t)$. The inverse operators and operational identities are managed in the Mikusinski field $\mathcal{F}_{-1}$:

$S_K = I / K, \qquad S_K \cdot K = I,$

with $I$ the unity. The operational calculus encodes fractional initial value problems as algebraic quotient equations, with solutions expressed via convolution-series generalizing the exponential/Mittag-Leffler structures (Alkandari et al., 2024, Luchko, 2021).

Prabhakar-type fractional derivatives, involving generalized three-parameter Mittag–Leffler functions, are encoded by convolution with kernels $$P_{B,\gamma} = x^{B-1} E_{\alpha,B}^{\gamma}(\delta x^{\alpha})$$, and the corresponding operator inverses $S_{B,\gamma} = P_{B,\gamma}^{-1}$ provide an operational calculus in the associated Mikusinski field $M_{-1}$. The operational rule for the $n$th-level Prabhakar derivative becomes

$$\mathcal{M}\left\{ {}_{0}\mathbf{D}^{(n)}_{\alpha, \mathbf{B}, \gamma, \delta} f \right\}= S_{B,\gamma}\mathcal{M}\{f\} - S_{B,\gamma}\sum_{k=0}^{M-1}c_k\,\mathcal{M}\left\{x^{\,B+S_n-k-1} E_{\alpha,B+S_n-k}^{\gamma(n-\theta_n)}(\delta x^\alpha)\right\},$$

with $\mathcal{M}$ the Mikusinski transform (Waheed et al., 24 Dec 2025, Alkandari et al., 2024).

3. Difference-Differential and q-Difference Structures

Mikusinski fields naturally admit difference structures, particularly $q$-difference and Mahler-type difference-differential calculi. For the $q$-shift $\tau_q(f)(t) = q f(qt)$, the automorphism $\sigma_q$ extends to $Q(C)$, and derivations are twisted accordingly:

$$\sigma_q(l) = ql,\qquad \sigma_q(s) = s/q, \qquad D = -s^2 \frac{d}{ds}, \qquad D \sigma_q = q \sigma_q D.$$

In Mahler-type difference fields, translations and dilations are captured by shift operators $h$ with $h * f(t) = f(t-1)$ and the associated automorphism $\sigma_d(h) = h^d$ (with $q=1/d$). The compatible derivation $D' = -h^{-1} d/ds$ satisfies $D' \sigma_d = d h^{d-1} \sigma_d D'$. These structures demonstrate that $Q(C)$ supports a uniform algebraic treatment for standard, $q$-difference, and Mahler difference-differential calculi (Nishioka, 16 Jun 2025).

Structure	Automorphism	Compatible Derivation	Commutator Relation
Classical	$T^\alpha$	$D = d/ds$	$D\,T^\alpha = T^\alpha D$
$q$-difference	$\sigma_q$	$D =-s^2 d/ds$	$D\,\sigma_q = q \sigma_q D$
Mahler	$\sigma_d$	$D' = -h^{-1} d/ds$	$D'\,\sigma_d = d h^{d-1}\sigma_d D'$

4. Noncommutative Mikusinski Calculus and Boundary Problems

A noncommutative extension localizes rings of regular boundary problems, forming a left ring of fractions $S^{-1}R$ over the differential-operator ring augmented with boundary data. Integro-differential algebras $(A, D, \int)$ with denote $D$ (differentiation) and $\int$ (Baxter operator) admit a module $\mathcal{F}$ of methorious hyperfunctions, including both classical and boundary-distributional elements. The generalized operator $s=(D, [_{\xi}])$ satisfies

$s \cdot f = D(f) + f(0) \delta_0,$

with boundary elements $\delta_{\xi}$ behaving as Dirac distributions. This extension incorporates both regular and singular boundary data in the operational calculus, enabling direct symbolic solution of boundary-value problems (Rosenkranz et al., 2012).

5. Algebraic Realizations and Formal Series

The operational algebra admits purely algebraic realization as modules over fields of convergent Laurent series $\mathcal{L}$, with actions compatible with integration ($t$), differentiation ($s=t^{-1}$), and functional shifts (e.g., $E_a f(t) = e^{a t} f(t)$ acts as $f(s-a)$). The quotient

$G(I) = M(I) / N(I),$

with $M(I)$ the module over $\mathcal{L}$ of Mikusinski functions and $N(I)$ the submodule representing constants and their images under differentiation, yields a model isomorphic to the space of finite-order Schwartz distributions, with operational rules mirroring Heaviside-Laplace calculus within the algebraic category (Lomadze, 2016). Extensions to functionals relevant to Bessel-type equations or more general special-function settings are managed via suitable group algebras and algebraic transforms (Bengochea et al., 2013).

6. Solution of Fractional and Functional Equations

In all the above settings, Mikusinski-type operational calculus reduces differential or fractional differential equations to algebraic relations in the convolution-quotient field. Initial or boundary value problems are handled via explicit inversion or partial-fraction decomposition, and explicit solutions are reconstructed in the time-domain using convolution series, often involving Mittag-Leffler or Prabhakar-type functions for fractional cases. The operational calculus for general fractional derivatives with Sonine kernels, $n$th-level Prabhakar derivatives, and Dzherbashian-Nersesian operators provides a systematic algorithmic route to closed-form solutions and supplies fundamental existence and uniqueness theorems based on the structure of the quotient field and convergence properties of the convolution expansions (Ahmad et al., 2023, Waheed et al., 24 Dec 2025, Alkandari et al., 2024).

7. Generalizations, Applications, and Structural Roles

Mikusinski-type operational calculi subsume the classical operational calculus, regular and singular fractional derivatives, functional equations of difference, $q$-difference, and Mahler-type, as well as equation classes of nonlocal and spectral type. They underpin modern studies in fractional PDEs, provide algebraic tools for the treatment of noncommutative boundary problems, and connect directly to the algebraic theory of formal series, special functions, and distribution theory. All operational relations—multiplication, inversion, partial fractions—translate to explicit convolution or hyperfunction formulas, giving uniform and rigorous foundations to a broad spectrum of analytic operational solution methods (Alkandari et al., 2024, Nishioka, 16 Jun 2025, Waheed et al., 24 Dec 2025, Rosenkranz et al., 2012, Lomadze, 2016).