MathDB

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)

Problem Statement