The generalized Bessel differential equation may also be obtained, aside from any physical or engineering motivations, by a purely mathematical argument, starting from the original Bessel differential equation and replacing the coefficients of the dependent variable and its derivative by polynomials of the independent variable; in this case the origin remains a regular singularity of the differential equation and the only other singularity is the point-at-infinity.
The solutions of the generalized Bessel differential equation around the regular singularity at the origin has (i) indices that are exponents of the leading power depending only on the order; (ii) recurrence relation for the coefficients of the power series expansion depending also on the degree.
The origin is a regular singularity of the generalized Bessel equation with the same indices (Section 2.1) as the original, leading to generalized Bessel functions whose series expansion differs from the original in the coefficients following the leading term.
The origin z = 0 is a regular singularity, and the only other singularity is at infinity; thus  the solution as a Frobenius series [5, 9] by the Fuchs theorem [6,10] converges in the whole finite complex z-plane
The origin is a regular singularity and the other singularity is at infinity, so the Frobenius-Fuchs method specifies power series solutions valid in the finite complex plane.