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We study Hawking radiation and wave scattering of charged scalar fields in acharged $C$-metric black hole background. The conformally invariant waveequation for charged scalar fields can be separated into radial and angularparts, each with five singularities. We show that the radial and angularequations can be respectively transformed into the general Heun equation. Weexplore exact solutions of the radial Heun equation in terms of the local Heunfunctions and connection coefficients. Exact behaviours of the asymptotic wavefunctions are determined without approximations. We apply the exact results toderive Hawking radiation, quasi-normal modes and superradiance. Sincequasinormal modes are significant for black holes through gravitional waves, wepresent the numeric results for quasinormal modes, and show the dependence uponthe $C$-metric parameters and the charge of scalar fields. The analyticexpressions of the solutions allow us to make fast numerical calculations ofhigh precision without restrictions on the model parameters.

Charged massless scalar fields in a charged $C$-metric black hole: Exact solutions, Hawking radiation and scattering of scalar waves