I'm interested in expressing the function:
$x\mathcal{L}_N^m (x^2 +y^2)$
in the form
$\sum_{N_i}\sum_{m_i} c_{N_i}^{m_i}\mathcal{L}_{N_i}^{m_i} (x^2 +y^2)$
where $c_{N_i}^{m_i}$ is a constant independent of $x,y$. Most of the identities I've looked up make it easy to do this for a function that looks more like
$f(x^2 +y^2)\mathcal{L}_N^m (x^2 +y^2)$
but this case is a little different. Any help is appreciated!