
#1
Nov1109, 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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