www.iaset.us editor@iaset.us 2 NEW SOLUTIONS OF ANGULAR TEUKOLSKY EQUATION VIA TRANSFORMATION TO HEUNS EQUATION WITH THE APPLICATION OF RATIONAL POLYNOMIAL OF AT MOST DEGREE 2 TOGETHER WITH AN INTEGRAL OPERATOR S. AKINBODE 1 & A. ANJORIN 2 1 Department of Mathematics and Statistics, East Tennessee State University, TN, U.S.A 2 Department of Mathematics, Lagos State University, Apapa Lagos, Nigeria ABSTRACT The perturbation equation of masseless fields for Kerr-de Sitter geometry are written in form of seperable equations as in [19] called the Angular Teukolsky equation. The Angular Teukolsky equation is converted to General Heun’s equation with singularities coinciding through some confluent process of one of five singularities. As in [17] and [18] rational polynomials of at most degree two are introduced. AMS Subject classification: 33XX. KEYWORDS: Heun Equation, Teukolsky Equation, Type-D Metrics, Polynomial Solutions 1. INTRODUCTION Teukolsky equations are the consequences of perturbation equation fer Kerr- de Sitter geometry with the separability of angular and radial parts respec- tively. Carter [1] was the first to discover that the scalar wave function is Separable other consideration is the 1 Spin electromagnetic field, gravita- tional perturbations and gravitino for the Kerr-deSitter class of geometry. The Teukolsky equation is applicable in he study of black holes in gen-eral. The solutions of the equation are in most cases expressed as series solutions of some specialized functions. This approach has been carried out by so many researchers say Teukolsky (1973), Breuer et all (1977), Frackerelland Crossman (1977), Leahy and Unruh (1979), Chakrabarti (1984),Siedel(1989), Suzuki et all (1989) just to mention but few.AlthoughTeukolskyequationhasfivesingularpointsoneirregularwithfourregularpoints.Bysome confluent process, these singular points are reduced to four coinciding with the singular points of Heun’ s equation. The objective of this work is to obtain polynomial solutions for the de- rived Tuekolsky equation through its conversion to Heun’ s equation through rational polynomials of degree at most 2. New solutions in terms of the ra- tional polynomials are obtained The paper is organized as follows; The first section deals with the intro- duction of Teukolsky equation, the second section deals with the derivation of Teukolsky using the work of [19], the third section has to do with the derivation of Angular Teukolsky and its conversion to Heun’ s equation and the fourth section has to do with Heun’ s differential equation and its trans- formation to hypergeometric differential equation via rational polynomials of at most degree two. International Journal of Applied Mathematics & Statistical Sciences (IJAMSS) ISSN(P): 2319-3972; ISSN(E): 2319-3980 Vol. 5, Issue 5, Aug - Sep 2016; 53-64 © IASET