Explicit solutions of Jacobi and Gauss differential equations by means of operators of fractional calculus

摘要

Judging by the remarkably large number of recent publications on fractional calculus and its applications in several widely diverse areas of mathematical, physical and engineering sciences, the current popularity and importance of the subject of fractional calculus cannot be overemphasized. Motivated by some of these potentially useful developments, many authors have recently demonstrated the usefulness of fractional calculus in the derivation of explicit particular solutions of a number of linear ordinary and partial differential equations of the second and higher orders. The main object of the present paper is to show how several interesting contributions on this subject, involving a certain class of ordinary differential equations associated with (for example) the celebrated Gauss and Jacobi differential equations, can be obtained (in a unified manner) by suitably applying some general theorems on explicit particular solutions of a family of linear ordinary fractional differintegral equations.