Exact Solutions to Sourceless Charged Massive Scalar Field Equation on Kerr-Newman Background

The separated radial part of a sourceless massive complex scalar field equation on the Kerr-Newman black hole background is shown to be a generalized spin-weighted spheroidal wave equation of imaginary number order. While the separated angular part is an ordinary spheroidal wave equation. General exact solutions in integral forms and in power series expansion as well as several special solutions with physical interest are given for the radial equation in the non-extreme case. In the extreme case, power series solution to the radial equation is briefly studied. Recurrence relations between coefficients in power series expansion of general solutions and connection between the radial equation are discussed in both cases.