Wigner numbers

J Chem Phys. 2019 Dec 28;151(24):244122. doi: 10.1063/1.5135721.


All reduced Wigner rotation matrix elements dM'M J(θ) can be evaluated very efficiently as a linear combination of either cos(Nθ) or sin(Nθ) terms as N runs in unit steps from either 0 or 12 to J. Exact, infinite-precision formulas are derived here for the Fourier coefficients in these dM'M J(θ) expressions by finding remarkable analytic solutions for the normalized eigenvectors of arbitrarily large matrices that represent the ĴY angular momentum operator in the basis of ĴZ eigenstates. The solutions involve collections of numbers Wm,n J for (m, n) = (J-M, J-N) ∈ [0, 2J] that satisfy the recursion relation (m+1)Wm+1,n J-2(J-n)Wm,n J+(2J-m+1)Wm-1,n J=0. These quantities, designated here as Wigner numbers, are proved to be integers that exhibit myriad intriguing mathematical properties, including various closed combinatorial formulas, (M, N) sum rules, three separate M-, N-, and J-recursion relations, and a large-J limiting differential equation whose applicable solutions are products of a polynomial and a Gaussian function in the variable z = -21/2(J + 1)-1/2M. Accordingly, the Wigner numbers constitute a new thread of mathematics extending outside the context of their immediate discovery. In the midst of the Wm,n J proofs, a class of previously unknown combinatorial summation identities is also found from Wigner number orthonormalization conditions.