Generating function for Laguerre polynomials (${\sum }_{n=0}^{\infty }{L}_n(x)h^n=(1-h{)}^{-1}{e}^{-xh/(1-h)}$)

The generating function holds for $|h|<1$ and fixed $x$. From Rodrigues' formula, $L_n(x)={\sum }_{j=0}^n\left(\genfrac{}{}{0ex}{}{n}{j}\right)(-x{)}^j/j!$. Sum first over $n\ge j$, using ${\sum }_{n\ge j}\left(\genfrac{}{}{0ex}{}{n}{j}\right)h^n=h^j/(1-h{)}^{j+1}$, and then over $j$ to obtain the exponential function. The absolute-value double series is finite for $|h|<1$, justifying the rearrangement. Alternatively, the weighted inner product and the Laplace transform of a Laguerre polynomial extract the coefficient $h^n$.