On a Generating Function for the Isotropic Basis Functions and Other Connected Results

Abstract

Recently isotropic basis functions of $N$ unit vector arguments were presented; these are of significant use in measuring the N-Point Correlation Functions (NPCFs) of galaxy clustering. Here we develop the generating function for these basis functions -- $i.e.$ that function which, expanded in a power series, has as its angular part the isotropic functions. We show that this can be developed using basic properties of the plane wave. A main use of the generating function is as an efficient route to obtaining the Cartesian basis expressions for the isotropic functions. We show that the methods here enable computing difficult overlap integrals of multiple spherical Bessel functions, and we also give related expansions of the Dirac Delta function into the isotropic basis. Finally, we outline how the Cartesian expressions for the isotropic basis functions might be used to enable a faster NPCF algorithm on the CPU.