Note on Legendre decomposition of the Pontryagin density in Kerr

Abstract

In arXiv:1406.0957v1 ("Scalar field excited around a rapidly rotating black hole in Chern-Simons modified gravity"), Konno and Takahashi have recently developed some analytical results for the scalar field about a Kerr black hole in the decoupling limit of dynamical Chern-Simons gravity. This involved a decomposition of the source (the Pontryagin density) in terms of Legendre polynomials. Here we give a two-line expression for this decomposition which simplifies their quadruple sum. Our expressions are rational polynomials multiplying Legendre functions of the second kind, or equivalently rational polynomials multiplying hypergeometric functions.