May 6, 2021

Continued Fractions, Attempt 3, Part 2: Defining Finite and Infinite Continued Fractions

1. Finite Continued Fractions

A finite continued fraction $\gamma$ is a finite sequence of pairs of complex numbers, that is, \[ \gamma=((a_0, b_0), (a_1, b_1), \ldots, (a_{n-1}, b_{n-1})) \] where $n\in\N$ and $a_k,b_k\in\C$ for any $k\in\{0,1,\ldots,n-1\}$. Each $a_k$ is called a partial denominator, and $b_k$ a partial numerator. We say $\gamma$ has length $n$, or write \[ \len(\gamma)=n. \] Note that the definition includes an empty sequence $()$, which has length $0$. We use $\cC$ to denote the set of all the finite continued fractions.

Define a projective transformation $\pi_{a,b}:\CP^1\to\CP^1$ by $\pi_{a,b}=\pi_{M(a,b)}$ where \[ \quad M(a,b)=\M{a}{b}{1}{0}\in\Mat(2,\C). \] $\pi_{a,b}$ is degenerate if and only if $b=0$. In the view of $\phy$ defined in Part 1, $\pi_{a,b}$ is represented in $\whC$ as \[ z\mapsto a+\frac{b}{z}. \]

The value of $\gamma$, denoted as $\lang\gamma\rang$, is defined by \[ \lang\gamma\rang=(\phy\circ\pi_{a_0,b_0}\circ\pi_{a_1,b_1}\circ\cdots\circ\pi_{a_{n-1},b_{n-1}})\left(\B{1}{0}\right). \] For simplicity, we write \[ \lang(a_0, b_0), (a_1, b_1), \ldots, (a_{n-1}, b_{n-1})\rang \] instead of $\lang((a_0, b_0), (a_1, b_1), \ldots, (a_{n-1}, b_{n-1}))\rang$. In particular, $\lang\rang=\infty$.

Remark. Traditionally, a continued fraction is a fraction of the form \[ a_0+\cfrac{b_0}{a_1+\cfrac{b_1}{\cdots+\cfrac{\cdots}{a_{n-2}+\cfrac{b_{n-2}}{a_{n-1}}}}}. \] This naive definition has several problems as follows.

  1. Is $\infty$ permitted as intermediate or final values?
  2. Are $a_k$’s and $b_k$’s can be $\infty$? If so, how to deal with $\infty/\infty$?
  3. Are $b_k$’s can be 0? If so, how to deal with $0/0$?

We have just given a rigorous definition of the value as $\lang(a_0, b_0), (a_1, b_1), \ldots, (a_{n-1}, b_{n-1})\rang$, clearly answering the above problems.

  1. $\infty$ is permitted as intermediate and final values.
  2. $a_k$’s and $b_k$’s cannot be $\infty$.
  3. $b_k$’s can be $0$. In the view of degenerate projective transformations, when $b_k=0$, $b_k/\cdots$ is always $0$, even if the denominator is also $0$.

2. Degenerateness of Finite Continued Fractions

Let $\gamma$ be a finite continued fraction $((a_0, b_0), (a_1, b_1), \ldots, (a_{n-1}, b_{n-1}))$. We say $\gamma$ is degenerate if and only if there exists $k\in\{0,1,\ldots,n-1\}$ such that $b_k=0$. The empty continued fraction $()$ is not degenerate.

Assume that $\gamma$ is degenerate. We have \[ \exists k_0 \forall l: b_{k_0}=0 \wedge(0\le l<k_0\rightarrow b_l\ne0). \] Define $\gamma'$ by \[ \gamma'=((a_0, b_0), (a_1, b_1), \ldots, (a_{k_0}, 1)), \] which is a non-degenerate finite continued fraction. Then, we have \[ \lang\gamma\rang=\lang\gamma'\rang. \] This means that, as far as values are of interest, we need to consider only non-degenerate continued fractions.

3. Limits of Sequences in $\whC$

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

4. Infinite Continued Fractions

An infinite continued fraction $\gamma$ is an infinite sequence of pairs of complex numbers, that is, \[ \gamma=((a_0, b_0), (a_1, b_1), (a_2, b_2), \ldots) \] where $a_k,b_k\in\C$ for any $k\in\N$. Each $a_k$ is called a partial denominator, and $b_k$ a partial numerator. We say $\gamma$ has length $\infty$, or write \[ \len(\gamma)=\infty. \] We use $\cC_\infty$ to denote the set of all the infinite continued fractions.

For any $n\in\N$, an $n$-th convergent $c_n$ of $\gamma$ is defined by \[ c_n=\lang(a_0,b_0),(a_1,b_1),\ldots,(a_{n-1},b_{n-1})\rang. \]

Let $\kappa=(c_n)_{n\in\N}$, which is a sequence in $\whC$. When $\kappa$ converges in $\whC$, the value of $\gamma$, denoted as $\lang\gamma\rang$, is defined by \[ \lang\gamma\rang=\lim_{n\to\infty}c_n. \]

5. Degenerateness of Infinite Continued Fractions

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

Assume that $\gamma$ is degenerate and let $c_n$ be an $n$-th convergent of $\gamma$. We have \[ \exists k_0 \forall l:(b_{k_0}=0)\wedge(0\le l<k_0\rightarrow b_l\ne0)\wedge(c_{k_0}=c_{k_0+1}=c_{k_0+2}=\cdots). \] Define $\gamma'$ by \[ \gamma'=((a_0,b_0),(a_1,b_1),\ldots,(a_{k_0},1)), \] which is a non-degenerate finite continued fraction. Then, we have \[ \lang\gamma\rang=\lang\gamma'\rang. \] This means that, as far as values are of interest, degenerate infinite continued fractions are essentially non-degenerate finite continued fractions (and thus always have values), and only non-degenerate infinite continued fractions are “truly” infinite.

Kenichi Kondo © 2001-2021