This document discusses solutions to the differential equation R(r) for Hawking radiation with vanishing mass M and angular frequency ω. It presents: 1) A transformation of the differential equation into Legendre form and the identification of regular singular points. 2) A power series solution and the derivation of a recurrence relation to determine the coefficients. 3) The identification of the Legendre polynomials as a solution and the first two polynomials are given as examples. 4) The derivation of a linearly independent logarithmic solution and the determination of its coefficients through the imposition of boundary conditions.