Spin-Matrix Polynomials and the Rotation Operator for Arbitrary Spin
Thomas A. Weber, Sarah Anne Williams · Journal of Mathematical Physics · 1965
A technique for the expansion of an arbitrary analytic function of a spin matrix is developed in terms of a complete set of polynomials based on the characteristic equations of the spin matrices. The expansion coefficients are determined by use of the ascending difference operator from the calculus of finite differences. The expansions are valid not only for functions of a spin matrix but also for functions of a complex variable. In the latter case the eigenvalues of the spin matrices are the zeros of the polynomials. In either case the expansion coefficients are the same although of course in the case of a spin matrix the series terminates. As an example the rotation operator for arbitrary spin is developed via its functional analog in terms of these polynomials, and the region of convergence of the series to the function is investigated.