I would like to rewrite the following series $$ \sum_{n=0}^\infty \frac{1}{n!}(\Delta^\epsilon)^n a_n , \qquad \Delta^\epsilon=\sum_{k=1}^\infty \epsilon^k b_k $$ As a series in $\epsilon$ $$ \sum_{n=0}^\infty c_n \epsilon^n $$ (i.e. I need to find the c_n).

Obviously, I can do this term-by-term. But the general case seems quite difficult. Any help would be greatly appreciated.