Approximation of analytic functions by Bessel functions of fractional order

Abstract

We solve the inhomogeneous Bessel differential equation $$x^2y''(x) + xy'(x) + (x^2 - \nu^2)y(x) = \sum^{\infty}_{m=0} a_mx^m$$ , where $\nu$ is a positive nonintegral number, and use this result for the approximation of analytic functions of a special type by the Bessel functions of fractional order.