Continued fractions seem to be somewhat a minor topic in modern mathematics, and the author thinks it’s one of reasons why he has always been feeling difficulty in finding literature which gives minimal and clean description of them. Therefore, he decided to create a starting point for that purpose.
Important notice: content of this article will be totally taken over by those of Part 5 and following articles.
Introduction
A continued fraction is, conceptually, a fraction of the form \[ a_0 + \cfrac{b_0}{a_1 + \cfrac{b_1}{a_2 + \cfrac{b_2}{\cdots}}}, \] where $a_n$ and $b_n$ are taken from a certain algebra depending on the context. We are going to give a precise definition and properties when this algebra is $\C$.
Finite Continued Fractions
Let $\whC$ denote $\C \cup \{\infty\}$ and $\C^\times$ denote $\C \setminus \{0\}$.
A finite continued fraction is a fraction represented by the bracket notation $[\,]$, which is defined by \[ [a_0,\underline{b_0},a_1,\underline{b_1},\ldots,a_{n-2},\underline{b_{n-2}},a_{n-1}]=a_0+\frac{b_0}{[a_1,\underline{b_1},\ldots,a_{n-2},\underline{b_{n-2}},a_{n-1}]}, \quad [a_{n-1}]=a_{n-1}, \] where $a_k\in\whC$ and $b_k\in\C^\times$ for all $k\in\{0,1,\ldots,n-1\}$. For example, \[ [a_0,\underline{b_0},a_1,\underline{b_1},a_2,\underline{b_2},a_3]=a_0+\cfrac{b_0}{a_1+\cfrac{b_1}{a_2+\cfrac{b_2}{a_3}}} \] is a finite continued fraction. Note that we regard $[a_0]=a_0$ as a valid finite continued fraction.
Infinite Continued Fractions
Let $a_n\in\whC$ and $b_n\in\C^\times$ for all $n\in\N$. Note that we include $0$ in $\N$.
An infinite continued fraction is a sequence $\kappa=\{c_n\}_{n\in\N}$ where $c_n$ is defined by \[ c_n=[a_0,\underline{b_0},a_1,\underline{b_1},\ldots,a_{n-1}], \quad c_0=\infty. \] Each term $c_n$ is called a convergent. For simplicity, we denote infinite continued fractions like \[ \kappa=[a_0,\underline{b_0},a_1,\underline{b_1},a_2,\underline{b_2},\ldots]=a_0+\cfrac{b_0}{a_1+\cfrac{b_1}{a_2+\cfrac{b_2}{\cdots}}}. \]
Remark. One of the reasons why we define $c_0$ to be $\infty$ is that we can always write \[ c_n=[a_0,\underline{b_0},a_1,\underline{b_1},\ldots,a_{n-1},\underline{b_{n-1}},\infty] \] and thus $c_0=[\infty]=\infty$. This will become more natural in a later section where we regard continued fractions as repetitions of Möbius transformations.
Simple Continued Fractions
A continued fraction, finite or infinite, is called simple when all the underlined elements in the bracket notation are 1. And such elements may be omitted from the notation. For example, \[ [a_0,a_1,a_2,a_3]=[a_0,\underline{1},a_1,\underline{1},a_2,\underline{1},a_3]=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3}}} \] and \[ [a_0,a_1,a_2,\ldots]=[a_0,\underline{1},a_1,\underline{1},a_2,\underline{1},\ldots]=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{\cdots}}} \] are simple continued fractions.
Remark. A widely used notation is $[a_0; a_1, a_2, \ldots, a_{n-1}]$, which is basically same as $[a_0, a_1, a_2, \ldots, a_{n-1}]$ in this article. The notation $[a_0,\underline{b_0},a_1,\underline{b_1},\ldots,a_{n-1}]$ is uncommon.
Non-degenerateness Condition
Let us consider a finite continued fraction $[a_0,\underline{b_0},a_1,\underline{b_1},\ldots,a_{n-2},\underline{b_{n-2}},a_{n-1}]$. If there exists $l\in\{0,1,\ldots,n-1\}$ such that $a_l=\infty$, then we have \[ \begin{aligned} [a_0,\underline{b_0},a_1,\underline{b_1},\ldots,a_{l-1}]&=[a_0,\underline{b_0},a_1,\underline{b_1},\ldots,a_{l-1},\underline{b_{l-1}},a_l] \\ &=[a_0,\underline{b_0},a_1,\underline{b_1},\ldots,a_{l-1},\underline{b_{l-1}},a_l,\underline{b_l},a_{l+1}] \\ &=\cdots \\ &=[a_0,\underline{b_0},a_1,\underline{b_1},\ldots,a_{l-1},\underline{b_{l-1}},a_l,\underline{b_l},a_{l+1},\ldots,\underline{b_{n-2}},a_{n-1}], \end{aligned} \] that is, $a_{l+1},a_{l+2},\ldots,a_{n-1}$ and $b_{l-1},b_l,\ldots,b_{n-2}$ does not affect the value at all. Similarly, for an infinite continued fraction with $a_l=\infty$, we have $c_l=c_{l+1}=c_{l+2}=\cdots$ and it is almost a finite continued fraction. We call these cases degenerate ones.
Thus, hereafter, we assume the non-degenerateness condition, which is of course defined as \[ a_n\ne\infty \] for all $n\in\N$.
Möbius Transformations and Convergents
Let $M_n$ be $\M{a_n}{b_n}{1}{0}\in\text{GL}_2(\C)$ and $m_n:\whC\rightarrow\whC$ the associated Möbius transformation to $M_n$, that is, \[ m_n(x)=a_n+\frac{b_n}{x}. \] Apparently, \[ c_n=(m_0\circ m_1\circ\cdots\circ m_{n-1})(\infty). \]
Theorem 1.1. We have $c_n=p_n/q_n$ where $p_n$ and $q_n$ are defined by the recurrence relation \[ \begin{aligned} \phantom{\text{(1.1a)}}&\hspace{4em} & p_{n+2}&=a_{n+1}p_{n+1}+b_np_n, & p_1&=a_0, & p_0&=1, & \hspace{4em}&\text{(1.1a)} \\ \phantom{\text{(1.1b)}}& & q_{n+2}&=a_{n+1}q_{n+1}+b_nq_n, & q_1&=1, & q_0&=0. & &\text{(1.1b)} \end{aligned} \]
Proof. Define $L_n$ by the recurrence relation \[ L_{n+1}=L_nM_n, \quad L_0=\M{1}{0}{0}{1}. \] Then $m_0\circ m_1\circ\cdots\circ m_{n-1}$ is the associated Möbius transformation to $L_n$. Therefore, if we write $L_n=\M{p_n}{r_n}{q_n}{s_n}$, we have $c_n=p_n/q_n$. On the other hand, we have \[ \begin{aligned} p_{n+1}&=a_np_n+r_n, & p_0&=1, \\ q_{n+1}&=a_nq_n+s_n, & q_0&=0, \\ r_{n+1}&=b_np_n, & r_0&=0, \\ s_{n+1}&=b_nq_n, & s_0&=1. \end{aligned} \] By eliminating $r_n$ and $s_n$, we get $L_{n+1}=\M{p_{n+1}}{b_np_n}{q_{n+1}}{b_nq_n}$ and the recurrence relation $\text{(1.1a)}$, $\text{(1.1b)}$.
Corollary 1.2. For any $n\in\N$ which satisfies $c_{n+1}\ne\infty$ and $c_{n+2}\ne\infty$, we have \[ c_{n+2}-c_{n+1}=(-1)^n\frac{b_nb_{n-1}\cdots b_0}{q_{n+2}q_{n+1}}. \]
Proof. If $c_{n+1}\ne\infty$ and $c_{n+2}\ne\infty$, we have \[ c_{n+2}-c_{n+1}=\frac{p_{n+2}}{q_{n+2}}-\frac{p_{n+1}}{q_{n+1}}=\frac{p_{n+2}q_{n+1}-p_{n+1}q_{n+2}}{q_{n+2}q_{n+1}}=\frac{\det L_{n+2}}{b_{n+1}q_{n+2}q_{n+1}} \] and \[ \det L_{n+2}=\det M_0 \det M_1 \cdots \det M_{n+1} = (-1)^n b_0b_1\cdots b_{n+1}. \]
Remark. Since there is no $n\in\N$ for which both $c_{n+1}$ and $c_{n+2}$ are $\infty$, it is tempting to extend the above formula for the cases where either $c_{n+1}$ or $c_{n+2}$ is $\infty$. But the difficulty is we must handle $+\infty$, $-\infty$, and $\infty$ as if they are different elements although they are identical in the view of $\whC$. Therefore, we restrict ourselves to the cases where both are finite.
Appendix
The first few terms of $\kappa=\{c_n\}_{n\in\N}$, written in the form of $p_n/q_n$, are \[ \begin{aligned} c_0&=\frac{1}{0}, \\ c_1&=\frac{a_0}{1}, \\ c_2&=\frac{a_1a_0+b_0}{a_1}, \\ c_3&=\frac{a_2a_1a_0+a_2b_0+b_1a_0}{a_2a_1+b_1}, \\ c_4&=\frac{a_3a_2a_1a_0+a_3a_2b_0+a_3b_1a_0+b_2a_1a_0+b_2b_0}{a_3a_2a_1+a_3b_1+b_2a_1}, \text{and} \\ c_5&=\frac{a_4a_3a_2a_1a_0+a_4a_3a_2b_0+a_4a_3b_1a_0+a_4b_2a_1a_0+a_4b_2b_0+b_3a_2a_1a_0+b_3a_2b_0+b_3b_1a_0}{a_4a_3a_2a_1+a_4a_3b_1+a_4b_2a_1+b_3a_2a_1+b_3b_1}. \end{aligned} \]
References
- [1] S. Hitotsumatsu, Introduction to Special Functions, Morikita, 1999.
Revisions
- August 20, 2020
- Stop revising; see Part 5 and following articles for renewed descriptions.
- July 26, 2020
- Simplify the introduction order of notions.
- July 16, 2020
- Describe finite continued fractions, introduce the bracket notation $[\,]$, and various minor improvements.
- May 6, 2020
- Various minor improvements.
- April 29, 2020
- Initial publication.