[seqfan] Re: Generating functions
William Keith
william.keith at gmail.com
Sat Dec 10 11:21:51 CET 2011
"It's not an inverse in the ring of polynomials, it's an inverse in the
> field of polynomials. 1-x-x^2 is exactly the inverse of the Fibonacci
> generating function in the field of polynomials over the integers."
>
> William, you lost me there. Did you mean "...in the set of all polynomials
> over the rationals" (the integers are not a field)? Otherwise, can you
> define
> a typical element of your field?
>
> I beg your pardon for obscurity. I meant the field of rational functions:
ratios of polynomials, or elements of the form P(x)/Q(x), where P and Q are
polynomials, Q != 0. 1/(1-x-x^2) is an element in this field, which has as
its power series expansion the Fibonacci series. The main point I was
trying to make was that while the ring of polynomials does not include a
general inversion operation, there is a setting in which the inverse you
were asking about is well defined.
Best,
William Keith
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