Express a quotient of polynomials as a continued fraction

Question

I have two polinomials P(x) and Q(x), with the degree of P larger than the degree of Q, both polynomials with known real coefficients. Is there a method to express P/Q as a continued fraction of x?

Answer 1

You have the division algorithm in polynomials, so:

$$p(x)=q(x)d_1(x)+r_1(x)\tag{1}$$ for some $d_1(x),r_1(x)$ with $\deg r_1(x)<\deg q(x).$

Then divide both sides of (1) by $q(x)$ and you get:

$$\frac{p(x)}{q(x)}=d_1(x)+\frac{r_1(x)}{q(x)}=d_1(x)+\dfrac1{\frac{q(x)}{r_1(x)}}$$

Now continue by applying the same process to $\frac{q(x)}{r_1(x)}.$ So we have:

$$q(x)=r_1(x)d_2(x)+r_2(x): \deg r_2<\deg r_1\\ \frac{q(x)}{r_1(x)} = d_2(x)+\dfrac1{\frac{r_1(x)}{r_2(x)}}$$

Keep doing this until the remainder $r_n(x)=0,$ and the process stops. It will eventually terminate because the degrees of the $r_i$ are strictly decreasing.

Then $$\frac{p(x)}{q(x)}=d_1(x)+\dfrac1{d_2(x)+\dfrac1{ \ddots+\frac1{d_n(x)}}}$$

This really is the same process as for rational numbers:

$$\begin{align}\frac{12}{5}&=2+\frac{2}{5}\\ &=2+\dfrac1{\frac{5}{2}}\\ &=2+\dfrac1{2+\frac12} \end{align}$$