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Point particles and Appell's solutions on the axis of Kerr black hole for arbitrary spin in terms of the Debye potentials

The Teukolsky master equation -- a fundamental equation for test fields of any spin, or perturbations, in type D spacetimes -- is classically treated in its separated form. Then the solutions representing even the simplest sources -- point particles -- are expressed in terms of series. The only known exception is a static particle (charge or mass) in the vicinity of Schwarzschild black hole. Here, we present a generalization of this result to a static point particle of arbitrary spin at the axis of Kerr black hole. A simple algebraic formula for the Debye potential from which all the NP components of the field under consideration can be generated is written down explicitly. Later, we focus on the electromagnetic field and employ the classic Appell's trick (moving the source into a complex space) to get so called electromagnetic magic field on the Kerr background. Thus the field of nontrivial extended yet spatially bounded source is obtained. We also show that a static electric point charge above the Kerr black hole induces, except an expected electric monopole, also a magnetic monopole charge on the black hole itself. This contribution has to be compensated. On a general level we discuss Teukolsky-Starobinsky identities in terms of the Debye potentials.