Infinite Degree Polynomials: Describing by Roots

[foxjwill]

Homework Statement

Is it possible to describe some infinite degree polynomials by their roots in a way analagous to finite degree polynomials?

Homework Equations

The Attempt at a Solution

I know that, since not all infinite degree polynomials have roots (e.g. the power series representation of e^x), it would not be possible to do so for all of them. But what about polynomials like the power series of sin(x)? I was thinking maybe

$$\prod^\infty_{n=0} \left ( x^2 - n^2\pi^2 \right )$$

 

[Hurkyl]

Is it possible to describe some infinite degree polynomials...

There's no such thing as an infinite degree polynomial. I presume you mean a power series.

If (the analtyic continuation) of your power series is actually meromorphic, then there is a general factorization theorem. See:

http://en.wikipedia.org/wiki/Infinite_product