Exponential Generating FunctionDate: 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/ |
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