It is written explicitly in wikipedia, https://en.wikipedia.org/wiki/Bell_polynomials#Generating_function, how one obtains a simple analytic expression for the Taylor series of the exponential of a function using Bell polynomials.
This formula gives schematically
$\exp^{f(x_1)-f(0)}=1+\sum^{\infty}_{n=1}\frac{x_1^n}{n!}\sum_{k=1}^{n}B_{n,k}(\partial_{x_1}f,\dots, \partial^{n-k+1}_{x_1}f)$.
It seems to me therefore that there should be a relatively simple way to write the analogous object for
$\exp^{f(x_1,x_2)-f(0,0)}=1+\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\frac{x^n_1}{n!}\frac{x^m_2}{m!}(\cdots)$,
or otherwise some other user- friendly way to write out this expansion. I am aware that one could do 1 variable expansion and then insert the second variable expansion into the Bell polynomials above, but it is not clear for me how to then simplify it further.
I need this because I am comparing with a distinct object which equates to $G[x_1,x_2]=\exp^{f(x_1,x_2)-f(0,0)}$ and I would like to match powers of $x_1,x_2$ for the taylor expansions of G and f.
Any such friendly expressions would be much appreciated.
Thanks for your time!