[b]6.[/b] 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

$F(t,x) = \sum_{n=0}^{\infty} \frac{P_{n}(x)}{n!}t^{n}.$

[b](S. 22)[/b]