[seqfan] Re: generation function ology question

[William Keith]

On Tue, Apr 17, 2012 at 4:33 AM, Peter Lawrence
<peterl95124 at sbcglobal.net>wrote:

>
> if G(x) is the generating function for sequence a(n), then is there a
> formula for the GF of the sequence a(K-n) ?
>
> thanks,
> Peter Lawrence.
>

Bearing in mind that these indices may not all be defined for a given
sequence:

If G(x) =\sum-{x=-\infty}^{\infty} a(n) x^n, then G(1/x) = \sum a(n) x^(-n)
= \sum a(-n) x^n.

In the latter expression, the coefficient of x^n is a(-n).  If you multiply
by x^K, you get

x^K G(1/x) = \sum a(-n) x^{K+n} = \sum a(K-n) x^n

so that the coefficient on x^n is the term a(K-n) of the sequence, which is
usually what is meant by the generating function.  A particular case of
this is when the sequence is finite of length L, so that X^L G(1/x)
reverses the sequence.  Sequences, especially polynomials, fixed under this
operation are symmetric in that sense.  Hope that helps.

William Keith