Computation of Jacobi functions of the second kind for use in nearside-farside scattering theory

The nearside-farside decomposition of a partial wave series is currently being used to understand the angular scattering of atom-diatom collision systems. In this theory, it is necessary to compute Jacobi functions of the second kind on the cut. These functions are denoted by Qn(α,β) (cos θ), where n, α, β, may be large positive integers. The Qn(α,β) (cos θ) can be computed from a three-term linear recurrence relation provided the initial values corresponding to n = 0 and 1, are known. We derive explicit formulas for Q0(α,β) (cos θ), Q1(α,β) (cos θ) in terms of elementary transcendental functions. A new generating function for Jacobi functions of nonintegral degree off the cut is obtained, a special case of which yields a generating function for Qn(α,β) (cos θ). This is used to check the numerical results, as is a Casoratian relation. We show that the recurrence for Qn(α,β) (cos θ) is stable in the forward direction with errors growing like O(n). We also present some numerics demonstrating the success of the method.