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Exponential Generating Function


Date: 05/06/2000 at 14:15:59
From: Kevin
Subject: Exponential Generating Functions

I'm trying to prove the following, but can't get to the final result:
 
Given the series 1, 1*3, 1*3*5, 1*3*5*7, ... prove that its 
exponential generating function is 1/sqrt(1-2*x).


Date: 05/06/2000 at 16:33:23
From: Doctor Anthony
Subject: Re: Exponential Generating Functions

Expand using the binomial theorem.

     (1 - 2x)^(-1/2) = 1 + (-1/2)(-2x) + (-1/2)(-3/2)(-2x)^2/2!
                        + (-1/2)(-3/2)(-5/2)(-2x)^3/3! + ...

For example, the term in x^3/3! has coefficient:

     (1)(3)(5)

and so this expansion is the exponential generating function for the 
series:

   1,  (1)(3),  (1)(3)(5), ...  and so on. 

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Sequences, Series

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