Wigner numbers.

All reduced Wigner rotation matrix elements d (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 d (θ) expressions by finding remarkable analytic solutions for the normalized eigenvectors of arbitrarily large matrices that represent the Ĵ angular momentum operator in the basis of Ĵ eigenstates. The solutions involve collections of numbers W for (m, n) = (J-M, J-N) ∈ [0, 2J] that satisfy the recursion relation (m+1)W -2(J-n)W +(2J-m+1)W =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 = -2(J + 1)M. Accordingly, the Wigner numbers constitute a new thread of mathematics extending outside the context of their immediate discovery. In the midst of the W proofs, a class of previously unknown combinatorial summation identities is also found from Wigner number orthonormalization conditions.