Re: closed form for generating function

From: David Moews (dmoews_at_xraysgi.ims.uconn.edu)
Date: 01/31/05 

Date: 31 Jan 2005 23:06:37 GMT

In article <1107210511.586737.93670@f14g2000cwb.googlegroups.com>,
hawkmoon269 <rson@new.rr.com> wrote:
|Could someone illustrate the derivation of the closed form for the
|following generating function?
|
|x / (1-nx)(1-x-x^2)
|
|I believe this would be a general form the convolution of the Fibonacci
|numbers and the powers of n. Thanks in advance.

If you want to find an expression for the coefficient of x^m in this function,
you can do so by expanding x/((1-nx)(1-x-x^2)) in partial fractions. However,
if you just want to compute the sum F_m + F_{m-1} n + ... + F_1 n^{m-1},
it would probably be easier to replace F_m by its exponential representation
(phi^m - (-phi)^(-m))/(phi + 1/phi) and then sum the geometric series. 

-- 
David Moews                                       dmoews@xraysgi.ims.uconn.edu

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