September 23, 2020

Continued Fractions, Part 6: Redefining Infinite Continued Fractions

This is the sixth article in the continued fractions series, following Part 5.

Convergence in $\whC$

Let $\sigma$ be an infinite sequence $(s_n)_{n\in\N}$ of elements of $\whC$. We say $\sigma$ converges to a finite value $\alpha\in\C$ if and only if \[ \forall\epsi\!\in\!\R_+\;\exists n_0\!\in\!\N\;\forall n\!\in\!\N: n>n_0\rightarrow(s_n\ne\infty\wedge|\alpha-s_n|<\epsi), \] where $\R_+=\{x\;|\;x\in\R\wedge x>0\}$. We represent this by writing \[ \lim_{n\to\infty}s_n=\alpha \] as usual. Also, we say $\sigma$ converges to $\infty$ if and only if \[ \forall r\!\in\!\R_+\;\exists n_0\!\in\!\N\;\forall n\!\in\!\N: n>n_0\rightarrow|s_n|>r, \] where we consider $|\infty|$ is greater than any real number. We represent this by writing \[ \lim_{n\to\infty}s_n=\infty \] as usual. Finally, we say $\sigma$ converges in $\whC$ if and only if $\sigma$ converges to a finite value or to $\infty$.

Infinite Continued Fractions

Let $\cC_\infty$ denote the set of all the infinite sequences of elements of $\C^2$, that is, \[ \cC_\infty=\{((a_0,b_0),(a_1,b_1),(a_2,b_2),\ldots)\,|\, (a_n,b_n)\in\C^2\;\text{for all $n\in\N$}\}. \] We call elements of $\cC_\infty$ infinite continued fractions.

Let $\gamma=((a_0,b_0),(a_1,b_1),(a_2,b_2),\ldots)\in\cC_\infty$ and define $\kappa=(c_n)_{n\in\N}$ by \[ c_n=[(a_0,b_0),(a_1,b_1),\ldots,(a_{n-1}, b_{n-1})], \quad c_0=[]. \] We say $\gamma$ has a value $\alpha$ in $\whC$ if and only if $\kappa$ converges in $\whC$ and \[ \alpha=\lim_{n\to\infty} c_n. \] For simplicity, we write \[ [(a_0,b_0),(a_1,b_1),(a_2,b_2),\ldots]=\lim_{n\to\infty} c_n. \] We call each element $c_n$ of $\kappa$ a convergent, and $\kappa$ the sequence of convergents.


We say an infinite continued fraction $\gamma=((a_0,b_0),(a_1,b_1),(a_2,b_2),\ldots)$ is degenerate if and only if there exists $k\in\N$ such that $b_k=0$.

Proposition 6.1. If $\gamma=((a_0,b_0),(a_1,b_1),(a_2,b_2),\ldots)$ is degenerate, say $b_k=0$, then $\gamma$ has a value in $\whC$ and \[ [(a_0,b_0),(a_1,b_1),(a_2,b_2),\ldots]=[(a_0,b_0),(a_1,b_1),\ldots,(a_k,b_k)]. \]

Proof. Define $\kappa=(c_n)_{n\in\N}$ by \[ c_n=[(a_0,b_0),(a_1,b_1),\ldots,(a_{n-1}, b_{n-1})], \quad c_0=[]. \] By Proposition 5.1 in Part 5, we have \[ c_{k+1}=c_{k+2}=c_{k+3}=\cdots \] and therefore \[ \lim_{n\to\infty}c_n=c_{k+1}. \]

Standard Embedding

We define the injection $\iota:\cC\to\cC_\infty$ by \[ \iota(((a_0,b_0),(a_1,b_1),\ldots,(a_{n-1},b_{n-1})))=((a_0,b_0),(a_1,b_1),\ldots,(a_{n-1},b_{n-1}),(a_n,b_n),(a_{n+1},b_{n+1}),\ldots) \] where \[ (a_n,b_n)=(0,1) \] and \[ (a_l,b_l)=(0,0) \] for all $l>n\;(l\in\N)$.

Proposition 6.2 For all $\gamma\in\cC$, we have \[ [\iota(\gamma)]=[\gamma]. \]

Proof. Writing $\gamma=((a_0,b_0),(a_1,b_1),\ldots,(a_{n-1},b_{n-1}))$, we have, by Proposition 6.1, \[ \begin{aligned} [\iota(\gamma)]&=[(a_0,b_0),(a_1,b_1),\ldots,(a_{n-1},b_{n-1}),(0,1),(0,0)] \\ &=(m_{a_0,b_0}\circ m_{a_1,b_1}\circ\cdots m_{a_{n-1},b_{n-1}})([(0,1),(0,0)]) \\ &=(m_{a_0,b_0}\circ m_{a_1,b_1}\circ\cdots m_{a_{n-1},b_{n-1}})(\infty) \\ &=[(a_0,b_0),(a_1,b_1),\ldots,(a_{n-1},b_{n-1})] \\ &=[\gamma]. \end{aligned} \]

Because of this proposition, we call $\iota$ the standard embedding of $\cC$ in $\cC_\infty$.

Kenichi Kondo © 2001-2021