New Aspects of Certain Special Functions with Applications. - Abdalla, Mohamed (Karshi Multidisciplinary International Scientific Journal, 2024)
Throughout the history of natural science, special functions (SFs) have been a powerful instrument in the solution of a wide variety of important problems in various fields such as physics, engineering, biology, medicine, economics, and finance. The reduction of any given applied problem to the evaluation of special functions has always been, and still is, looked on as indicating a strong penetration into the essence of the problem. Almost all the familiar special functions have arisen from a wide diversity of applied problems, so the study of their properties and their applications has engaged not only mathematics but also physicists, astronomers, engineers, and other specialists. Furthermore, fractional calculus of special functions, which perform fractional differentiation or integration of functions, is gaining popularity due to its numerous scientific, technological, and engineering applications. So, the current review article dives into recent mathematical findings on generalizations of special functions and polynomials related to various fractional calculus operators (FCOs). Also emphasized is their importance and extensive utility in dealing with the most well-known topics: integral transformations, initial value problems, and kinetic equations. More precisely, we discuss analytic properties and numerical exemplifications of extensions Beta and hypergeometric functions associated with fractional calculus operators. Moreover, some developments and applications of orthogonal matrix polynomials, such as the generalized Bessel matrix polynomials and the generalized Jacobi matrix polynomials, have been considered. Furthermore, novel generalizations of fractional kinetic equations involving certain special functions and their solutions using the various integral transforms have been shown. Finally, some important points that can be suitable to be future works are summarized.