# Moment problem

1. Nov 11, 2009

### zetafunction

given the set of moments $$m_k$$

obtained from the known measure w(x) $$\int _{-\infty}^{\infty}dx w(x) x^{k} = m_k$$

now if we define the moment generating function $$f(z)= \sum_{k=0}^{\infty} m_{k}(-1)^{k}z^{k}$$ , which have the integral representation

$$f(z)= \int _{-\infty}^{\infty} \frac{w(x)}{1+xz }$$

my question is , how could i expand f(z) knowing the moment m_k in the form of a continued fraction

$$f(z) = a_0 (z)+ \frac{1 \mid}{\mid a_1 (z)} + \frac{1 \mid}{\mid a_2 (z)} + \frac{1 \mid}{\mid a_3 (z)}.$$

where the a_n are Polynomials of degree 1 .

for example for the case of exponential integral , how did euler manage to get the continued fraction ?? thanks.