# Infinite polynomials

1. Feb 5, 2008

### foxjwill

1. The problem statement, all variables and given/known data
Is it possible to describe some infinite degree polynomials by their roots in a way analagous to finite degree polynomials?

2. Relevant equations

3. 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

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

2. Feb 5, 2008

### Hurkyl

Staff Emeritus
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