In the near future, cosmology will enter the wide and deep galaxy survey era, enabling high-precision studies of the large-scale structure of the universe in three dimensions. To test cosmological models and determine their parameters accurately, it is necessary to use data with exact theoretical expectations expressed in observational parameter space (angles and redshift). The data-driven, galaxy number count fluctuations on redshift shells can be used to build correlation functions $\xi (\theta ,{z}_{1},{z}_{2})$ on and between shells to probe the baryonic acoustic oscillations and distance-redshift distortions, as well as gravitational lensing and other relativistic effects. To obtain a numerical estimation of $\xi (\theta ,{z}_{1},{z}_{2})$ from a cosmological model, it is typical to use either a closed form derived from a tripolar spherical expansion or to compute the power spectrum ${C}_{{\ell }}({z}_{1},{z}_{2})$ and perform a Legendre polynomial ${P}_{{\ell }}(\cos \theta )$ expansion. Here, we present a new derivation of a $\xi (\theta ,{z}_{1},{z}_{2})$ closed form using the spherical harmonic expansion and proceeding to an infinite sum over multipoles thanks to an addition theorem. We demonstrate that this new expression is perfectly compatible with the existing closed forms but is simpler to establish and manipulate. We provide formulas for the leading density and redshift-space contributions, but also show how Doppler-like and lensing terms can be easily included in this formalism. We have implemented and made publicly available software for computing those correlations efficiently, without any Limber approximation, and validated this software with the CLASSgal code. It is available at https://gitlab.in2p3.fr/campagne/AngPow.