Special Functions Associated with Fractional Calculus

Dear Colleagues,

Many important functions in applied sciences are defined via improper integrals or series (or infinite products). Those functions are generally called special functions. Special functions contain a very old branch of mathematics. For example, trigonometric functions have been studied for over a thousand years, due mainly to their numerous applications in astronomy. Nonetheless, the origins of their unified and rather complete theory date back to the nineteenth century. From an application point of view, special functions such as important mathematical tools, due to their remarkable properties, are designated so based on their usefulness for the applied scientists and engineers—as Paul Tur´an once remarked, special functions would be more appropriately labeled useful functions. Various special functions, such as Bessel and all cylindrical functions; the Gauss, Kummer, confluent, and generalized hypergeometric functions; the classical orthogonal polynomials; the incomplete Gamma and Beta functions and error functions; the Airy, Whittaker functions; etc., will provide solutions to integer-order differential equations and systems, used as mathematical models. However, there has recently been an increasing interest in and widely extended use of differential equations and systems of fractional order (that is, of arbitrary order) as better models of phenomena of physics, engineering, automatization, biology and biomedicine, chemistry, earth science, economics, nature, and so on. Today, new unified presentation and extensive development of special functions associated with fractional calculus are necessary tools related to the theory of differentiation and integration of arbitrary order (i.e., fractional calculus) and to fractional-order (or multiorder) differential and integral equations.

This Special Issue is to provide a multidisciplinary forum of discussion in diverse branches of mathematics and statistics but also physics, engineering, automatization, biology and biomedicine, chemistry, earth science, economics, nature, and so on. This issue will accept high-quality articles containing original research results and survey articles of exceptional merit. Subject matters should be strongly related to special functions involving fractional calculus. The main objective of this Special Issue is to highlight the importance of fundamental results and techniques of the theory of fractional calculus and let the readers of this issue know about the possibilities of this branch of mathematics. Potential topics include but are not limited to:

• Fractional calculus;
• Sequence and series in functional analysis;
• Generalized fractional calculus and applications;
• Fractional differential equations;
• Fractional derivatives and special functions;
• Various special functions related to generalized fractional calculus;
• Special functions related to fractional (non-integer) order control systems and equations;
• Applications of fractional calculus in mechanics;
• Applications of fractional calculus in physics;
• Special functions arising in the fractional diffusion-wave equations;
• Operational method in fractional calculus;
• Fractional integral inequalities and their q-analogues;
• Inequalities involving the fractional integral operators;
• Applications of inequalities for classical and fractional differential equations.

Prof. Dr. Ravi P. Agarwal
Prof. Dr. Praveen Agarwal
Prof. Dr. Shilpi Jain
Guest Editors

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