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Recursion Relation

Let $ z_\ell(x)$ be either solution or a linear combination of the two. $ x$ is a complex scalar independent variable (in practice, $ x =
kr$ ). Then

$\displaystyle z_{\ell + 1}(x) = \frac{2 \ell + 1}{x} z_\ell(x) - z_{\ell - 1}(x).$ (13.19)

This relation is stable for increasing $ \ell$ for $ z_\ell = n_\ell$ . It is stable for decreasing $ \ell$ for $ z_\ell = j_\ell$ . For that reason it is unstable in both directions for $ h^\pm_\ell$ (defined below). How would you make it? See Abramowitz and Stegun, Handbook of Mathmatical Functions for discussion of recursive algorithm and definition of power series expansions.

Robert G. Brown 2017-07-11