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Moment problem

by zetafunction
Tags: moment
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Nov11-09, 12:31 PM
P: 399
given the set of moments [tex] m_k [/tex]

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

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

[tex] f(z)= \int _{-\infty}^{\infty} \frac{w(x)}{1+xz } [/tex]

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

[tex] 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)}. [/tex]

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.
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