Analytic solutions of the Teukolsky equation and their properties

The analytical solutions reported in our previous paper are given as series of hypergeometric or Coulomb wave functions. Using them, we can get the Teukolsky functions analytically in a desired accuracy. For the computation, the deep understanding of their properties is necessary. We summarize the main result: The relative normalization between the solutions with a spin weight s and -s is given analytically using the Teukolsky-Starobinsky (T-S) identities. By examining the asymptotic behavior of our solution and combined with the T-S identities and the Wronskian, we found nontrivial identities between the sums of coefficients of the series. These identities will serve to make various expression in simpler forms and also become a powerful tool to test the accuracy of the computation. As an application, we investigated the absorption rate and the evaporation rate of black hole and obtain interesting analytic results.