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

TL;DR

This paper develops a generating function for isotropic basis functions used in galaxy clustering analysis, enabling efficient computation of overlap integrals and faster algorithms for N-Point Correlation Functions.

Contribution

It introduces a generating function for isotropic basis functions, facilitating the derivation of Cartesian expressions and improving computational methods for NPCFs.

Findings

Derived the generating function using properties of plane waves.

Enabled computation of complex overlap integrals of spherical Bessel functions.

Outlined potential for faster NPCF algorithms on CPUs.

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.