Legendre polynomial recurrence relation
= Legendre polynomial recurrence relation
{c}
The three-term recurrence
$$
(2\ell+1)xP_\ell(x)
=(\ell+1)P_{\ell+1}(x)+\ell P_{\ell-1}(x)
$$
turns multiplication by $x$ into coupling between adjacent Legendre multipoles.