Suppose I have the following:
$$xe^{xz}A(x,z)$$
where
$$A(x,z)=\sum_{n=0}^{\infty}\frac{a_n(z)}{n!}x^n$$
and $a_n$ is a polynomial in $z$
Taking the Cauchy product (since $e^{xz}=\sum_{n=0}^{\infty}\frac{z^n}{n!}x^n) $ gives
$$x\sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{a_k(z)z^{n-k}}{k!(n-k)!}x^n=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\binom{n}{k}\frac{a_k(z)z^{n-k}}{n!}x^{n+1}$$
I need to reindex with respect to $n$ since I want a power series that has terms $x^n$, so I think this yields
$$ \sum_{n=1}^{\infty}\sum_{k=0}^{n-1}\binom{n-1}{k}\frac{a_k(z)z^{n-1-k}}{(n-1)!}x^{n} $$
Would this be the correct reindexing?
Edit: I changed the denominator to $(n-1)!$ from $n!$ since I think I missed it...