I am reading a paper on combinatorics. It mentions some generating functions are rational series and others are algebraic series. I do not understand the difference, can someone help?
EDIT $1$: The paper is http://fr.arxiv.org/pdf/0810.4387v3. The very first page, under introduction, the very first paragraph, about half way through.
It's rational if it satisfies a linear equation with polynomial coefficients (that is, if it is a quotient of polynomials; that is, if it is a rational function). It's algebraic if it satisfies a polynomial equation with polynomial coefficients.
For example: the generating function $$f(x)=2+3x+5x^2+9x^3+17x^4+33x^5+\cdots$$ is a rational series because it satisfies the linear equation $$(1-3x+2x^2)f(x)-(2-3x)=0$$ which is another way to say it is the rational function $$f(x)={2-3x\over1-3x+2x^2}$$
The generating function $$g(x)=1+x+2x^2+5x^3+14x^4+42x^5+\cdots$$ where the coefficient of $x^n$ is the $n$th Catalan number, $$C_n={1\over n+1}{2n\choose n}$$ is an algebraic series because it satisfies the polynomial equation $$x(g(x))^2-g(x)+1=0$$