Generalized special functions such as the Aleph-function, Srivastava–Daoust function, Fox–H function, and Mittag–Leffler type functions have emerged as natural tools in the study of fractional differential equations and anomalous transport processes. These functions extend the expressive power of classical hypergeometric functions and allow the compact representation of solutions that cannot be expressed in terms of Bessel or Legendre functions alone. In particular, their appearance in fractional calculus has been instrumental for modeling memory and hereditary effects in viscoelastic materials, anomalous diffusion in complex or porous media, and non-Fourier heat conduction at the micro- and nano-scale.

A promising application domain is heat and mass transfer in heterogeneous or fractal media, where Fourier’s law is no longer adequate. The Fox–H and Aleph functions, for example, provide exact representations of Green’s functions for generalized diffusion equations with distributed-order derivatives. This makes them especially relevant in energy engineering (heat transfer in advanced composites, porous catalytic reactors) and environmental science (groundwater transport, pollutant diffusion). Their ability to interpolate between classical Gaussian diffusion and heavy-tailed Lévy flights makes them powerful for bridging theoretical models and experimental data.

In physics, these generalized functions also play a role in quantum mechanics and statistical mechanics, particularly in systems with non-standard potentials or non-extensive entropy formulations. The Srivastava–Daoust function, for instance, appears in exact solutions of certain multidimensional Schrödinger equations and in evaluating partition functions under generalized statistics. Similarly, Fox–H functions have been used to describe wave propagation in dispersive media, spectral densities in random matrix theory, and relaxation phenomena in complex systems. Their unifying property—reducing to simpler polynomials or hypergeometric functions under specific parameter limits—makes them attractive as analytical tools in both perturbative and non-perturbative regimes.

From a computational perspective, generalized special functions also provide advantages in applied mathematics and engineering modeling. They offer closed-form representations of solutions that would otherwise require costly numerical simulation, thus enabling more efficient algorithms for boundary value problems, integral transforms, and symbolic computation. With the current rise of machine learning and operator-theoretic methods in applied mathematics, having such flexible function classes is particularly valuable: they serve both as analytical benchmarks and as functional bases for data-driven modeling of PDEs. In this sense, the applications of generalized special functions are expanding beyond traditional mathematical physics into emerging areas of complex systems, network science, and computational engineering.

References

Mathai, A. M., Saxena, R. K., & Haubold, H. J. (2010). The H-Function: Theory and Applications. Springer.

Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Elsevier.

Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press. (review-style monograph with broad applications overview)