Recursive relation for power series of the exponential

Question

Is there a recursive relation, like one for geometric series

$$G_n=1+x+x^2+...+x^n$$ $$G_{n+1}=1+xG_n$$

For the taylor expansion of the exponential function?

$$e^x_n=\sum\limits_{i=0}^n\frac{x^i}{i!}$$

Omitting of course the trivial case

$$e^x_{n+1}=e_n^x+\frac{x^{n+1}}{(n+1)!}$$

Or perhaps expressing the entire power expansion by means of other functions?

Answer 1

You might like the following integral representation

$$e^x_{n+1}=\int_0^xe^x_n(y)dy+1,$$

but probably you are looking for something different.