Statistics of Quantum Numbers for Non-Equivalent Fermions in Single-j Shells

This work addresses closed-form expressions for the distributions P(M) of the magnetic quantum numbers M and Q(J) of total angular momentum J for non-equivalent fermions in single-j orbits. Such quantities play an important role in both nuclear and atomic physics, through the shell models. Using irreducible representations of the rotation group, different kinds of formulas are presented, involving multinomial coefficients, generalized Pascal triangle coefficients, or hypergeometric functions. Special cases are discussed, and the connections between P(M) (and therefore Q(J)) and mathematical functions such as elementary symmetric, cyclotomic, and Jacobi polynomials are outlined.