It is known that
$\sum^{\infty}_{n=0}z^nL_n(x)=\frac{1}{1-z}e^{-xz/(1-z)}$
where $L_n(x)$ is the Laguerre polynomial.
It there any neat way of expressing the following term:
$\sum^{\infty}_{n=0}z^{n+m}L_{n+m}(x)$
where the index of the Laguerre polynomial and $z$ become $(n+m)$, and $m$ is a given integer?