Let G G be a group of linear transformations of R n {R^n} and H k ( G ) {H_k}(G) the vector space of spherical harmonics invariant under G G . The Pálya function is the formal power series Σ k ≥ 0 t k dim ⁡ H k ( G ) {\Sigma _{k \geq 0}}{t^k}\dim {H_k}(G) . In this paper, after classifying all closed subgroups of O ( 4 ) O(4) , we compute the Pólya functions for these groups. These functions have recently proved to be of interest in quantum mechanics and elementary particle physics.