Prove that continued fractions are fractional linear transformations

Question

Consider the continuous fractions defined inductively by

$a_1=1+\frac{1}{x}, a_2=1+\cfrac{1}{1+\frac{1}{x}}, a_3=1+\cfrac{1}{1+\cfrac{1}{1+\frac{1}{x}}}, ...$

Prove that each $a_n$ is a fractional linear transformation, that is

$a_n=\frac{b_nx+c_n}{g_nx+h_n}$

and find the coefficients $b_n, c_n, g_n, h_n$. Then prove that

$\lim_{n \rightarrow \infty}\frac{b_nx+c_n}{g_nx+h_n}=\frac{bx+c}{gx+h}$

and find $b, c, g, h$.

Please help, I have no clue where to start.

Answer 1

Let $(F_n)_{n\ge0}$ be the Fibonacci sequence defined by $F_0=0$, $F_1=1$, and $F_{n+1}=F_n+F_{n-1}$. Then it is an easy induction to show that $$a_n=\frac {F_{n+ 2}x+F_{n+1}}{F_{n+1}x+F_n}=\frac {\frac{F_{n+ 2}}{ F_{n+1}}x+1}{x+\frac{ F_n}{ F_{ n+1}}}$$ Now, since it is well-known that $$\lim\limits_{n\to\infty}\dfrac{F_{ n+1}}{F_n}=\varphi=\frac{1+\sqrt5}{2}$$ We conclude that $$\lim_{n\to\infty}a_n=\frac{\varphi x+1}{ x+1/\varphi}=\varphi$$ Which is independant of $x$.

Answer 2

Hint: Think Fibonacci. $$ 1 + \frac{1}{\frac{bx+c}{gx+h}} = \frac{\frac{(b+g)x + (c+g)}{gx + h}}{\frac{bx+c}{gx+h}} = \frac{(b+g)x + (c+g)}{bx + c} $$