  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.
