A survey of fractional-calculus approaches to the solutions of the Bessel differential equation of general order

In a remarkably large number of recent works, one can find the emphasis upon (and demonstrations of) the usefulness of fractional-calculus operators in the derivation of (explicit) particular solutions of significantly general families of linear ordinary and partial differential equations of the second and higher orders. The main object of this presentation is to survey some earlier investigations of this simple fractional-calculus approach to the solutions of the classical Bessel differential equation of general order and to show how it would lead naturally to several interesting consequences which include (for example) an alternative derivation of the complete power-series solutions obtainable usually by the Frobenius method. The underlying analysis presented here is based chiefly upon some of the general theorems on (explicit) particular solutions of a certain family of linear ordinary fractional differintegral equations with polynomial coefficients.