So suppose that the equation $$f(x,y)=\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\alpha_{ij}(x-c_1)^i(y-c_2)^j$$ holds in some open set containing the point $c=(c_1,c_2)$. Assume the $f$ can be differentiated to arbitrarily high orders by simply differentiating the series termwise. Show $$\alpha_{ij}=\frac{\binom{k}{i}}{k!}D_1^iD_2^jf(c_1,c_2),$$ where $k=i+j$.
I am completely lost. I've tried a few things that aren't worth writing down. I would assume that we use the generalized Taylor's theorem, but as I said I cannot get anywhere with it. Any help is much appreciated, thank you!