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

$$z_{\ell + 1}(x) = \frac{2 \ell + 1}{x} z_\ell(x) - z_{\ell - 1}(x).$$ (11.86)

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.