SCHWEITZER 1957 Problem 6

Let $f(x)$ be an arbitrary function, differentiable infinitely many times. Then the $n$ th derivative of $f(e^{x})$ has the form $\frac{d^{n}}{dx^{n}}f(e^{x})= \sum_{k=0}^{n} a_{kn}e^{kx}f^{(k)}(e^{x})$ ( $n=0,1,2,\dots$ ).

From the coefficients $a_{kn}$ compose the sequence of polynomials $P_{n}(x)= \sum_{k=0}^{n} a_{kn}x^{k}$ ( $n=0,1,2,\dots$ )

and find a closed form for the function $F(t,x) = \sum_{n=0}^{\infty} \frac{P_{n}(x)}{n!}t^{n}.$ (S. 22)