On the Divergence in the General Sense of $q$-Continued Fraction on the Unit Circle

TL;DR

This paper demonstrates that within a specific class of $q$-continued fractions, including famous examples like Rogers-Ramanujan, there are uncountably many points on the unit circle where these fractions diverge in the general sense, impacting their convergence understanding.

Contribution

It establishes the existence of uncountably many divergence points on the unit circle for a class of $q$-continued fractions, including notable examples like Rogers-Ramanujan.

Findings

Uncountably many divergence points on the unit circle.

Includes key continued fractions like Rogers-Ramanujan.

Implications for convergence of other $q$-continued fractions.

Abstract

We show, for each $q$-continued fraction $G(q)$ in a certain class of continued fractions, that there is an uncountable set of points on the unit circle at which $G(q)$ diverges in the general sense. This class includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fraction. We discuss the implications of our theorems for the general convergence of other $q$-continued fractions, for example the G\"{o}llnitz-Gordon continued fraction, on the unit circle.

Tables

Table 1. Table 1.

$G\left(q\right)$	$K\left(q\right)$	$S_1\left(q\right)$	$S_2\left(q\right)$	$S_3\left(q\right)$
$\eta$	$1/5$	$1/8$	$1/2$	$1/3$
$(s,d)$	$(1,5)$	$(1,8)$	$(1,8)$	$(1,6)$
$H\left(q\right)$	$\frac{2\exp \left(2\pi ir/5\right)}{1+\sqrt{5}}$	$\frac{1}{\sqrt{2}\exp \left(-\pi ir/4\right)}$	$\frac{1}{\left(1+\sqrt{2}\right)\exp \left(\pi ir\right)}$	$\frac{1}{2\exp \left(4\pi ir/3\right)}$
$k$	$1$	$2$	$2$	$1$
$f_1,\cdots ,f_k$	$x$	$q{x}^2$, $qx+q^2{x}^2$	$q{x}^2+q^2{x}^4$,$q^4{x}^4$	$x+x^2$
$a_{km}$	$1$	$2$	$1$	$2$
$P_{km-1}$	$1$	$2$	$3$	$1$
$Q_{km-2}$	$0$	$1$	$1$	$0$
$Q_{km-1}$	$q^{\left(m-1\right)/5}$	$ϵq^{\left(m^2-1\right)/8}$	$q^{\left(m-1\right)/2}$	$q^{\left(m-1\right)/3}$
$P_{km-2}$	$q^{\left(1-m\right)/5}$	$ϵq^{{\left(m-1\right)}^2/8}$	$q^{\left(m+1\right)/2}$	$q^{\left(2m+1\right)/3}$

Equations

$$K(q):=1\text{lower}5.83333pt+\frac{q}{1}\text{lower}5.83333pt+\frac{q^2}{1}\text{lower}5.83333pt+\frac{q^3}{1}\text{lower}5.83333pt+\text{lower}5.83333pt\cdots .$$

$$K_n(q):=1+\frac{q}{1}\text{lower}5.83333pt+\frac{q^2}{1}\text{lower}5.83333pt+\frac{q^3}{1}\text{lower}5.83333pt+\text{lower}5.83333pt\cdots \text{lower}5.83333pt+\frac{q^n}{1}.$$

$$\underset{j\to \infty }{\lim }{K}_{2j+1}(q)$$

$$\underset{j\to \infty }{\lim }{K}_{2j}(q)$$

$$K(q)=\lambda q^{(1-\lambda \sigma m)/5}K(\lambda ),$$

$$S$$

$$\begin{gathered} t=[0,2,2^{2^2},2^{2^{2^3}},\cdots ]= \\ 0.484848484848484848484848484848484848484848484848484848484 \\ 84848484848484848484849277885083112437522992318812011\cdots \end{gathered}$$

$$S_1(q):=1+\frac{q}{1}\text{lower}5.83333pt+\frac{q+q^2}{1}\text{lower}5.83333pt+\frac{q^3}{1}\text{lower}5.83333pt+\frac{q^2+q^4}{1}\text{lower}5.83333pt+\text{lower}5.83333pt\cdots ,$$

$$S_2(q):=1+\frac{q+q^2}{1}\text{lower}5.83333pt+\frac{q^4}{1}\text{lower}5.83333pt+\frac{q^3+q^6}{1}\text{lower}5.83333pt+\frac{q^8}{1}\text{lower}5.83333pt+\text{lower}5.83333pt\cdots ,$$

$$S_3(q):=1+\frac{q+q^2}{1}\text{lower}5.83333pt+\frac{q^2+q^4}{1}\text{lower}5.83333pt+\frac{q^3+q^6}{1}\text{lower}5.83333pt+\frac{q^4+q^8}{1}\text{lower}5.83333pt+\text{lower}5.83333pt\cdots .$$

$$M=b_0+\frac{a_1}{b_1+\frac{a_2}{b_2+\frac{a_3}{b_3+\text{lower}5.83333pt\cdots }}}$$

$$S_n(w)=\frac{A_n+w{A}_{n-1}}{B_n+w{B}_{n-1}}.$$

$$d(w,z)=\frac{|z-w|}{\sqrt{1+|w{|}^2}\sqrt{1+|z{|}^2}}$$

$$d(w,\infty )=\frac{1}{\sqrt{1+|w{|}^2}}.$$

$$\underset{n\to \infty }{\lim }{S}_n(v_n)=\underset{n\to \infty }{\lim }{S}_n(w_n)=f.$$

$$G(q):=b_0(q)+$$

$$:=g_0(q^0)+$$

$$+$$

$$\text{lower}5.83333pt\cdots \text{lower}5.83333pt+\frac{f_k(q^{n-1})}{g_0(q^n)}\text{lower}5.83333pt+$$

$$a_{nk+s}=a_{nk+s}(q)=f_s(q^n),$$

$$GG(q):=1+q+\frac{q^2}{1+q^3}\text{lower}5.83333pt+\frac{q^4}{1+q^5}\text{lower}5.83333pt+\frac{q^6}{1+q^7}\text{lower}5.83333pt+\text{lower}5.83333pt\cdots ,$$

$$P_n$$

$$Q_n$$

$$P_n{Q}_{n-1}-P_{n-1}{Q}_{n-1}$$

$$a_{jmk+r}$$

$$b_{jmk+r}$$

$$q=\exp \left(2\pi ir/m\right)$$

$$H\left(\exp \left(2\pi ir/m\right)\right)$$

$$a_{km}(q)$$

$$Q_{km-2}(q)$$

$$|Q_{km-1}(q)|$$

$$H\left(\exp \left(2\pi ir/s\right)\right)=H_a$$

$$H(q)=\frac{q^{(\lambda \sigma m)/5}}{\lambda K(\lambda )},$$

$$H(q)=\frac{2\exp (2\pi ir/5)}{1+\sqrt{5}}.$$

Full text

We discuss the implications of our theorems for the general convergence of other $q$-continued fractions, for example the Göllnitz-Gordon continued fraction, on the unit circle.

The second author’s research supported in part by a Trjitzinsky Fellowship.

1. Introduction

In [2], we made a detailed study of the convergence behaviour of the famous Rogers-Ramanujan continued fraction $K(q)$, where

[TABLE]

It is an easy consequence of Worpitzky’s Theorem (see [8], pp. 35–36) that $R(q)$ converges to a value in $\hat{\mathbb{C}}$ for any $q$ inside the unit circle.

Theorem 1**.**

(Worpitzky) Let the continued fraction $K_{n=1}^{\infty}a_{n}/1$ be such that $|a_{n}|\leq 1/4$ for $n\geq 1$. Then $K_{n=1}^{\infty}a_{n}/1$ converges. All approximants of the continued fraction lie in the disc $|w|<1/2$ and the value of the continued fraction is in the disk $|w|\leq 1/2$.

Suppose $|q|>1$. For $n\geq 1$, define

[TABLE]

Then

[TABLE]

This was stated by Ramanujan without proof and proved by Andrews, Berndt, Jacobson and Lamphere in 1992 [1].

This leaves the question of convergence on the unit circle. The convergence behaviour at roots of unity was investigated by Schur, who showed in [11] that if $q$ is a primitive $m$-th root of unity, where $m\equiv 0$ $(\text{mod}\,\,5)$, then $K(q)$ diverges and if $q$ is a primitive $m$-th root of unity, $m\not\equiv 0(\text{mod}\,\,5)$, then $K(q)$ converges and

[TABLE]

where $\lambda=\left(\frac{m}{5}\right)$ (the Legendre symbol) and $\sigma$ is the least positive residue of $m\,(\text{mod}\,\,5)$. Note that $K(1)=\phi=(\sqrt{5}+1)/2$, and $K(-1)=1/\phi$.

Remark: Schur’s result was essentially proved by Ramanujan, probably earlier than Schur (see [9], p.383). However, he made a calculational error (see [6], p.56).

There remains the question of whether the Rogers-Ramanujan continued fraction converges or diverges at a point on the unit circle which is not a root of unity. The chief difficulty in trying to apply the usual convergence/divergence tests stems from the facts that the Rogers-Ramanujan continued fraction converges at a set of points that is dense on the unit circle and diverges at another such dense set. This is clear from the result of Schur above.

This question about convergence on the unit circle at points which were not roots of unity remained unanswered until our paper, [2], where we showed the existence of an uncountable set of points on the unit circle at which the Rogers-Ramanujan continued fraction diverged.

To discuss this topic we use the following notation. Let the regular continued fraction expansion of any irrational $t\in(0,1)$ be denoted by $t=[0,e_{1}(t),e_{2}(t),\cdots]$. Let the $i$-th approximant of this continued fraction expansion be denoted by $c_{i}(t)/d_{i}(t)$. We will sometimes write $e_{i}$ for $e_{i}(t)$, $c_{i}$ for $c_{i}(t)$ etc, if there is no danger of ambiguity. Let $\phi=(\sqrt{5}+1)/2$. In [2], we proved the following theorem.

Theorem 2**.**

[2]** Let

[TABLE]

Then $S$ is an uncountable set of measure zero and, if $t\in S$ and $y=\exp(2\pi it)$, then the Rogers-Ramanujan continued fraction diverges at $y$.

We were also able to give explicit examples of points $y$ on the unit circle at which $K(y)$ diverges.

Corollary 1**.**

Let $t$ be the number with continued fraction expansion equal $[0,e_{1},e_{2},\cdots]$, where $e_{i}$ is the integer consisting of a tower of $i$ twos with an $i$ an top.

[TABLE]

If $y=\exp(2\pi it)$ then $K(y)$ diverges.

We were also able to show the existence of an uncountable set of points on the unit circle at which $R(q)$ diverges in the general sense (see below for the definition of general convergence) and to give explicit examples of such points (The point $y$ of Corollary 1 is such a point, for example).

In [3] we generalised Theorem 2 to a wider class of $q$-continued fractions, a class which includes the Rogers-Ramanujan continued fraction and the three “Ramanujan-Selberg” continued fractions studied by Zhang in [13]:

[TABLE]

[TABLE]

and

[TABLE]

These continued fractions were first studied by Ramanujan [9]. As a corollary to our theorem in [3], we were able to show, for each of the continued fractions above, the existence of an uncountable set of points on the unit circle at which the continued fraction diverged.

In this present paper we extend our result in [2] on the divergence in the general sense of the Rogers-Ramanujan continued fraction on the unit circle to a wider class of $q$-continued fractions, a class which includes $K(q)$, $S_{1}(q)$, $S_{2}(q)$ and $S_{3}(q)$. We show that each of these $q$-continued fractions diverges in the general sense at an uncountable set of points on the unit circle.

2. Divergence in the General Sense of $q$-Continued Fractions

on the Unit Circle

In [7], Jacobsen revolutionised the subject of the convergence of continued fractions by introducing the concept of general convergence. General convergence is defined, see [8], as follows.

Let the $n$-th approximant of the continued fraction

[TABLE]

be denoted by $A_{n}/B_{n}$ and let

[TABLE]

Define the chordal metric $d$ on $\hat{\mathbb{C}}$ by

[TABLE]

when $w$ and $z$ are both finite, and

[TABLE]

Definition: The continued fraction $M$ is said to converge generally to $f\in\hat{\mathbb{C}}$ if there exist sequences $\{v_{n}\}$, $\{w_{n}\}\subset\hat{\mathbb{C}}$ such that $\liminf d(v_{n},w_{n})>0$ and

[TABLE]

Remark: Jacobson shows in [7] that, if a continued fraction converges in the general sense, then the limit is unique.

The idea of general convergence is of great significance because classical convergence implies general convergence (take $v_{n}=0$ and $w_{n}=\infty$, for all $n$), but the converse does not necessarily hold. General convergence is a natural extension of the concept of classical convergence for continued fractions.

We consider continued fractions of the form

[TABLE]

where $f_{s}(x),g_{s-1}(x)\in\mathbb{Z}[q][x]$, for $1\leq s\leq k$. Thus, for $n\geq 0$ and $1\leq s\leq k$,

[TABLE]

Many well-known $q$-continued fractions, including the Rogers-Ramanujan continued fraction, the three Ramanujan-Selberg continued fractions studied by Zhang in [13], and the Göllnitz-Gordon continued fraction,

[TABLE]

have the form of the continued fraction at (2.2), with $k$ at most 2. It seems natural to consider a class of continued fractions which, in a sense, contains all of the above continued fractions.

For the remainder of the paper $P_{n}(q)/$ $Q_{n}(q)$ denotes the $n$-th approximant of $G(q)$, $P_{n}/Q_{n}$ if there is no danger of ambiguity. For later use, we recall some basic facts about continued fractions. It is well known (see, for example, [8], p.9) that the $P_{n}$’s and $Q_{n}$’s satisfy the following recurrence relations.

[TABLE]

It is also well known (see also [8], p.9) that, for $n\geq 1$,

[TABLE]

Condition 2.3 also implies that if $q$ is a primitive $m$-th root of unity then $G(q)$ is periodic with period $m\,k$. Indeed, if $q$ is a primitive $m$-th root of unity and $j\geq 0$,

[TABLE]

We now assume certain facts about the approximants of $G(q)$, and the convergence behaviour of $G(q)$, at certain roots of unity.

We assume that there is a positive integer $d$ and an integer $s\in\{1,2,$ $\ldots,$ $d\}$, such that if $m\equiv s\mod{d}$ and $(r,m)=1$, then

[TABLE]

This integer $s$ will be referred to frequently in what follows.

We further assume that if $G(q)$ converges at $q=\exp\left(2\pi ir/m\right)$, a primitive $m$-th root of unity, then $G(q)$ converges at any $q^{\prime}=\exp\left(2\pi ir^{\prime}/m^{\prime}\right)$, a primitive $m^{\prime}$-th root of unity, where $m\equiv m^{\prime}(\text{mod }d)$ and $r\equiv r^{\prime}(\text{mod }d)$.

We also assume that there exists $\eta\in\mathbb{Q}$ such that if $H(q):=q^{\eta}/G(q)$ and $G(q)$ converges at $q=\exp\left(2\pi ir/m\right)$ then

[TABLE]

with $r^{\prime}$ and $m^{\prime}$ as above. Note that the above condition implies that $H(q)$ takes only finitely many values at roots of unity. Let these values be denoted $H_{1}$, $H_{2}$,$\ldots,H_{N_{G}}$.

We assume that for all $m\equiv s$ (mod $d$) that there are integers $K_{0}$, $K_{1}$, $K_{2}$, $K_{3}$ and $K_{4}$, depending only on $s$, such that

[TABLE]

Here $k$ is the positive integer in the definition of the continued fraction $G(q)$ at (2.2).

Finally, it is also assumed that there exists $r\not=u\in\{0,1,\ldots,d-1\}$, such that

[TABLE]

for some $a$, $b\in\{1,\ldots,N_{G}\}$.

It may be instructive at this point to show how these abstract conditions above apply to a particular continued fraction. We let $G(q)=K(q)$.

If we compare the continued fractions at (1.1) and (2.2), it is clear that we can take $k=1$, $g_{0}(x)\equiv 1$ and $f_{1}(x)\equiv x$ (giving $b_{n}(q)=g_{0}(q^{n})=1$ and $a_{n}(q)=f_{1}(q^{n})=q^{n}$).

From Schur’s paper [11] (or see Table 1, which contains the relevant information from [11]) we can take $d=5$ and $s=1$ and if $q$ is a primitive $m$-th root of unity, $m\equiv 1\mod{5}$, then $K(q)$ converges, giving Condition 2.8 above.

If we set $\eta=1/5$ and set $H(q)=q^{1/5}/K(q)$, we have from (1.2) that, if $q$ is a primitive $m$-th root of unity, $m\not\equiv 0\mod{5}$, then

[TABLE]

where $\lambda=\left(\frac{m}{5}\right)$ (the Legendre symbol) and $\sigma$ is the least positive residue of $m\,(\text{mod}\,\,5)$. It follows that $H(q)$ can take only ten possible values at roots of unity.

It is also clear from (2.12) that Conditions 2.9 and 2.11 are satisfied, since if $m\equiv 1\mod{5}$ (so that $\lambda=\sigma=1$ and $K(1)=(1+\sqrt{5})/2$ ) and $q=\exp\left(2\pi ir/m\right)$, with $(r,m)=1$, then

[TABLE]

If $q$ is a primitive $m$-th root of unity, it follows from (1.1) and (2.10) that $K_{0}=a_{m}(q)=q^{m}=1$. It follows from Schur’s paper [11] (or, once again, from Table 1) that $K_{1}=P_{m-1}(q)=1$, $K_{2}=Q_{m-2}(q)=0$ and $K_{3}=P_{m-2}(q)Q_{m-1}(q)=q^{(1-m)/5}q^{(m-1)/5}=1=K_{4}$. Thus Condition 2.10 is satisfied.

From the paper of Zhang [13], each of $S_{1}(q)$, $S_{2}(q)$ and $S_{3}(q)$ also satisfy a set of conditions of the form set out in (2.8) to (2.11). The relevant details are found in Table 1.

As before, let the regular continued fraction expansion of an irrational $t\in(0,1)$ be denoted by $[0,e_{1},e_{2},\ldots]$ and let the $n$-th approximant of this continued fraction be denoted by $c_{n}/d_{n}$. We prove the following theorem.

Theorem 3**.**

Let $G(q)$ have the form given by (2.2) and satisfy conditions (2.3) and (2.8) – (2.11).

There exists an integer $N^{\prime}$ and a strictly increasing function $\gamma:\mathbb{N}\to\mathbb{N}$ such that if $t$ is any irrational in $(0,1)$ for which there exist two subsequences of approximants $\{c_{f_{n}}/d_{f_{n}}\}$ and $\{c_{g_{n}}/d_{g_{n}}\}$ satisfying

[TABLE]

*and *

[TABLE]

for all $n$, where $h_{n}=f_{n}$ or $g_{n}$. Then $H(\exp(2\pi it))$ does not converge generally.

Let $S^{\diamond}$ denote the set of all $t\in(0,1)$ satisfying (2.13) and (2.14) and set

[TABLE]

Then $Y_{G}$ is an uncountable set of measure zero.

We show, as a corollary to this theorem, for each of the continued fractions $K(q)$, $S_{1}(q)$, $S_{2}(q)$ and $S_{3}(q)$, that there exists an uncountable set of points on the unit circle at which the continued fraction does not converge generally.

The main idea of the proof will be to show that there exist points $y$ on the unit circle for which there exist two sequences of positive integers, $\{m_{i}\}$ and $\{n_{i}\}$, such that the subsequences of approximants to $H(y)$, $\{P_{n_{i}}/Q_{n_{i}}\}$ and $\{P_{n_{i}-1}/Q_{n_{i}-1}\}$ each tend to the same limit, $L_{1}$ say, and the subsequences $\{P_{m_{i}}/Q_{m_{i}}\}$ and $\{P_{m_{i}-1}/Q_{m_{i}-1}\}$ each tend to the same limit $L_{2}\not=L_{1}$. This is done by constructing real numbers $t$ in the interval $(0,1)$ whose continued fraction expansions have a certain rapid convergence behavior and then setting $y=\exp(2\pi it)$. In addition, it is shown that the sequences $\{Q_{n_{i}}/Q_{n_{i}-1}\}$ and $\{Q_{m_{i}}/Q_{m_{i}-1}\}$ are bounded from above, for $i$ sufficiently large. These two conditions are then shown to imply that $H(q)$ does not converge generally at $y$.

We first give some technical lemmas. The proofs are not given if the results are well known. Our aim is to estimate $P_{i}(q)$ and $Q_{i}(q)$ for sequences of $i$’s in certain arithmetic progressions. We use matrix notation since the proofs are simpler.

Lemma 1**.**

Let $G(q)$ be as in (2.2). There exist strictly increasing sequences of positive integers $\{\kappa_{n}\}$ and $\{\nu_{n}\}$ such that if $x$ and $y$ are any two points on the unit circle then, for all integers $n\geq 0$,

[TABLE]

and

[TABLE]

Proof.

Let $\{f_{n}(q)\}$ be any sequence of polynomials in $\mathbb{Z}[q]$. Suppose $f_{n}(q)=\sum_{i=0}^{M_{n}}\gamma_{i}q^{i}$, where the $\gamma_{i}$’s are in $\mathbb{Z}$. Then

[TABLE]

Now set $\delta_{n}=\max\left\{\,\,\sum_{i=1}^{M_{n}}i\,|\gamma_{i}|,\,\,\,1,\,\,\delta_{n-1}+1\right\}$. Inequality (1) follows by setting $f_{n}(q)=Q_{n}(q)$ and $\delta_{n}=\kappa_{n}$. The result for (2.17) follows similarly. ∎

With $\kappa_{n}$ and $\nu_{n}$ as in the above lemma, define, for each $n\geq 1$,

[TABLE]

This function will be used later in the proof of Theorem 3.

Lemma 2**.**

$($[12]**, p. 238$)$ For $n\geq 0$,

[TABLE]

Proof.

This follows, by induction, from the recurrence relations (2.5). ∎

We now assume that $q$ is a primitive $m$-th root of unity, $q=\exp(2\pi in/m)$, where $(n,m)=1$, $m\equiv s\mod{d}$ and either $n\equiv r\mod{d}$ and $n\equiv u\mod{d}$, where $r$, $s$ and $u$ are as in condition (2.11).

Lemma 3**.**

For $j\geq 1$ and $1\leq r\leq km$,

[TABLE]

For $j\geq 1$ and $0\leq r\leq km-1$,

[TABLE]

Proof.

By Lemma 2 and the periodicity of the $a_{i}$’s and/or the $b_{i}$’s noted at (2.7), we have that

[TABLE]

Statement (2.20) then follows from the facts that $P_{km}=b_{km}P_{km-1}+a_{km}P_{km-2}$ and $Q_{km}=b_{km}Q_{km-1}+a_{km}Q_{km-2}$ and Lemma 2. Statement (2.21) is an immediate consequence of (2.20). ∎

Remark: It is clear from (2.21), that if $G(q)$ converges then $Q_{km-1}\not=0$, since otherwise $Q_{jkm-1}=0$ for $j\geq 1$.

Define

[TABLE]

Equation (2.6) implies that

[TABLE]

Let $T$ denote the trace of $M$ and $D$ its determinant. In light of (2.23) and (2.10) it is clear that $T$ and $D$ are both integers that depend only on $s$. From this it is clear that

[TABLE]

and that $K_{5}$ also depends only on $s$. The eigenvalues of $M$ are

[TABLE]

The corresponding eigenvectors are

[TABLE]

and

[TABLE]

As shown above, if $G(q)$ converges, then $Q_{km-1}\not=0$, and this justifies taking $x_{2}=y_{2}=1$. Note for later use that $|x_{1}|$, $|y_{1}|$, $\lambda_{1}$ and $\lambda_{2}$ depend only on $s$. This follows from (2.10) and (2.24).

Lemma 4**.**

Let the eigenvalues of

[TABLE]

be $\lambda_{1}$ and $\lambda_{2}$. If $G(q)$ converges then $\lambda_{1}=\lambda_{2}$ or $|\lambda_{1}|\not=|\lambda_{2}|$.

Proof.

Since $a_{i}\not=0$ for $i\geq 1$, it follows from (2.6), that $\text{Det}(M)\not=0$, so that neither of the eigenvalues is zero.

Suppose $\lambda_{1}\not=\lambda_{2}$ but $|\lambda_{1}|=|\lambda_{2}|$. In this case it is clear from (2.25) and (2.26) that x and y are linearly independent. For $r\in\{0,1,2,\dots,km-1\}$, suppose that $(P_{r},Q_{r})^{T}=p_{r}$ x$+q_{r}$ y, for some $p_{r}$, $q_{r}\in\mathbb{C}$. Then it follows from (2.21), (2.26) and (2.27) that

[TABLE]

By some simple algebraic manipulation,

[TABLE]

The right hand side does not converge as $j\to\infty$, unless $p_{r}=0$ or $q_{r}=0$, for each $r$.

Since we are considering the case where no $a_{i}=0$, then $a_{1}\not=0$, $(P_{0},Q_{0})\not=\gamma(P_{1},Q_{1})$, for any $\gamma\in\mathbb{C}$. Hence $p_{0}q_{1}-p_{1}q_{0}\not=0$.

We first consider the case $p_{0}=0$. Then $\lim_{j\to\infty}$ $P_{jkm}$ $/Q_{jkm}$ $=$ $y_{1}$. Since $p_{0}=0$, it follows from the remark above that $p_{1}\not=0$, and then it must be that $q_{1}=0$ and $\lim_{j\to\infty}P_{jkm+1}/Q_{jkm+1}=x_{1}\not=y_{1}$, which is a contradiction.

On the other hand, if $p_{0}\not=0$, then $q_{0}=0$ and so $q_{1}\not=0$ which necessitates $p_{1}=0$, and a similar contradiction follows. This completes the proof. ∎

Remarks: 1) The eigenvalues for the Rogers-Ramanujan continued fraction and the Ramanujan-Selberg continued fractions are non-zero and distinct.

2. It follows similarly from (2.29), (2.26) and (2.27), that, in the case $|\lambda_{1}|\not=|\lambda_{2}|$,

[TABLE]

For later use we evaluate $G(q)$ when $\lambda_{1}=\lambda_{2}$. In this case $T^{2}-4\,D=0$. This equation implies

[TABLE]

This in turn means that $P_{km-1}\not=a_{km}Q_{km-2}$, or else $P_{km-2}=0$ and (2.21) gives that $P_{jkm-2}=0$ for $j\geq 1$, implying that $G(q)=0$, contradicting our assumption.

For ease of notation we write $P_{km-1}=a$, $Q_{km-1}=c$ and $a_{km}Q_{km-2}=d$. Then it follows from Lemma 3 and induction that

[TABLE]

From this and (2.21) it follows that, for $0\leq r\leq km-1$,

[TABLE]

and that

[TABLE]

This holds whether or not $2\,c\,P_{r}-(a-d)Q_{r}=0$, for any $r\in\{0,1,\ldots,km-1\}$.

Note that (2.10) implies that $|G(q)|$ depends only on $s$.

In the following lemmas the cases of equal and unequal eigenvalues are considered separately, with Lemmas 5 to 7 dealing with the case of equal eigenvalues.

Define $G_{n}(q):=P_{n}(q)/Q_{n}(q)$ and $H_{n}(q):=x^{\eta}/G_{n}(q)$, where $\eta$ is as defined at (2.9).

For the case of equal eigenvalues, it follows from (2.25) that $T=P_{km-1}+a_{km}Q_{km-2}\not=0$. Note also that the conditions at (2.10) imply that $P_{km-1}-a_{km}Q_{km-2}=K_{1}-K_{0}\,K_{2}$, a fixed integer depending only on $s$.

In the following lemmas a sequence of positive integers $N_{1},\ldots,N_{10}$ and a sequence of constants is defined. These integers and constants will depend only on the constants $K_{0}$, $K_{1}$, $K_{2}$ and $K_{3}$ described at (2.10). As such, they will depend only on $s$. To avoid repetition throughout the lemmas, we state here that these integers are chosen to satisfy $N_{1}<N_{3}<N_{4}<N_{7}<N_{8}$ and $N_{2}<N_{5}<N_{6}<N_{9}<N_{10}$. For Lemmas 5 to 7, we assume that $\lambda_{1}=\lambda_{2}$.

Lemma 5**.**

There exist positive constants $D_{1}$, $D_{2}$, $D_{3}$, $D_{4}$, $D_{5}$ and $D_{6}$, each depending only on $s$, such that if $j\geq 1$, then

[TABLE]

*There exists a positive integer $N_{1}$, depending only on $s$, such that if $j\geq N_{1}$, then *

[TABLE]

*and *

[TABLE]

Proof.

To prove (2.34), we first equate entries at (2.32), using the fact that $\lambda_{1}=\lambda_{2}=(P_{km-1}+a_{km}Q_{km-2})/2$.

[TABLE]

(2.34) follows upon setting $D_{1}=|\lambda_{1}|$ and $D_{2}=|Q_{km-1}/\lambda_{1}|$, recalling the conditions at (2.10) and the facts noted at the end of (2.27).

Note for later use that, since $M$ has determinant equal to a non-zero integer and has two equal eigenvalues, then $D_{1}\geq 1$ so that $\lim_{j\to\infty}|Q_{jkm-1}|=\infty$. Inequality (2.36) then implies $\lim_{j\to\infty}|Q_{jkm-1}|=\infty$ also.

Statement (2.35) follows similarly from comparing corresponding matrix elements at (2.32), namely,

[TABLE]

Set

[TABLE]

Take $N_{1}$ large enough so that $2|\lambda_{1}|/N_{1}<|a_{km}Q_{km-2}-P_{km-1}|$ and then set

[TABLE]

Statement (2.36) follows from (2.34) and (2.35), by taking $D_{5}=D_{2}/D_{4}$ and $D_{6}=D_{2}/D_{3}$. ∎

Lemma 6**.**

There exists positive constants $D_{7}$, $D_{8}$, $D_{9}$, $D_{10}$ and $D_{11}$ and positive integers $N_{3}$ and $N_{4}$, each depending only on $s$, such that, if $j\geq 1$, then

[TABLE]

if $j\geq N_{3}$, then

[TABLE]

and if $j\geq N_{4}$ and $n=jkm-1$ or $jkm-2$ , then

[TABLE]

Proof.

Equations (2.32) and (2.33) give that

[TABLE]

Set

[TABLE]

Note that $D_{7}\not=0$, since $D_{7}=|\lambda_{1}/Q_{km-1}|\not=0$. Next, from (2.32) and (2.33) we find that

[TABLE]

Choose $N_{3}$ such that $|(P_{km-1}+a_{km}Q_{km-2})/N_{3}|<|(P_{km-1}-a_{km}Q_{km-2})|$ and set

[TABLE]

and

[TABLE]

Note that neither $D_{8}$ or $D_{9}$ is zero, since $\lambda_{1}=(P_{km-1}+a_{km}Q_{km-2})/2\not=0$ and $P_{km-1}-a_{km}Q_{km-2}\not=0$ from the remark following (2.31).

Let $n=jkm-1$ or $jkm-2$ and set $M^{{}^{\prime}}=\max\{D_{7},D_{9}\}$ and $m^{{}^{\prime}}=\min\{D_{7},D_{8}\}$. Choose $N_{4}$ such that $|G(q)|>M^{{}^{\prime}}/N_{4}$ (Recall that $|G(q)|\not=0$ and is constant for fixed $s$). Let $j\geq N_{4}$. Then

[TABLE]

Then

[TABLE]

By the definition of $M^{\prime}$ and the choice of $N_{4}$, it follows that

[TABLE]

Set

[TABLE]

The constants $D_{10}$ and $D_{11}$ depend only on $s$, since $|G(q)|$, $m^{\prime}$, $M^{\prime}$ and $N_{4}$ depend only on $s$ . ∎

Lemma 7**.**

Let $y$ be another point on the unit circle. There exist positive constants $D_{13}$, $D_{14}$ and $D_{15}$ and positive integers $N_{7}$ and $N_{8}$, each depending only on $s$, such that if $j\geq N_{7}$ and $n=jkm-1$ or $jkm-2$, and

[TABLE]

then

[TABLE]

if $j\geq N_{8}$ and the angle between $q$ and $y$ (measured from the origin) is less than $\pi/(2|\eta|)$, then

[TABLE]

and

[TABLE]

Proof.

Let

[TABLE]

Set $D_{13}^{\prime}=3+|G(q)|+D_{12}/N_{7}$ and set $D_{13}=2D_{13}^{\prime}/D_{2}^{\prime}$. Choose $N_{8}$ such that

[TABLE]

From the fact that $D_{1}\geq 1$ together with (2.34) and (2.35), it follows that

[TABLE]

Let $j\geq N_{7}$. Then $|Q_{n}|-1/2>D_{2}^{\prime}\,j-1/2\geq D_{2}^{\prime}\,j/2$ and

[TABLE]

Here we have used (2.37), (2.38), the bounds on $\epsilon_{1}$ and $\epsilon_{2}$ and the inequality relating $|Q_{n}|$ and $j$ above.

Similarly, if $j\geq N_{8}$, then

[TABLE]

Here we have used (2.40) and the fact that the angle between $q$ and $y$ (measured from the origin) is less than $\pi/(2|\eta|)$ implies that $|y^{\eta}-q^{\eta}|\leq 2|\eta||q-y|$ (This last inequality follows since the stated bound implies $(q/y)^{\eta}$ lies in the first or fourth quadrant and the fact that in these quadrants, chordal distance from 1 is less than arc distance, which in turn is less than twice the chordal distance). From (2.37) and (2.38), it follows that

[TABLE]

Set $D_{14}=\max\{4/|G(q)|,\,\,4|\eta|/|G(q)|\}$ and $D_{15}=4\,D_{13}/|G(q)|^{2}$.

Statement (2.42) follows from (2.41) and (2.39). ∎

In the following three lemmas we assume $|\lambda_{1}|>|\lambda_{2}|$.

Lemma 8**.**

There exist positive constants $C_{1}$, $C_{2}$, $C_{3}$, $C_{4}$, $C_{5}$, $C_{6}$ and $C_{7}$, and a positive integer $N_{2}$, each depending only on $s$, such that, if $j\geq 1$, then

[TABLE]

and if $j\geq N_{2}$, then

[TABLE]

and

[TABLE]

Proof.

Let $\lambda_{1}$, $\lambda_{2}$, $x_{1}$ and $y_{1}$ be as defined at (2.25), (2.26) and (2.27). Then

[TABLE]

From Lemma 3 it follows that

[TABLE]

Thus

[TABLE]

Statement (2.45) follows with $C_{1}=|\lambda_{1}|$ and

[TABLE]

Note that, since $M$ has a non-zero integral determinant and $|\lambda_{1}|>|\lambda_{2}|$, $C_{1}=|\lambda_{1}|>1$.

Similarly,

[TABLE]

Choose $N_{2}$ large enough so that

[TABLE]

and then take

[TABLE]

and

[TABLE]

Note that equation (2.27) and the fact that none of $a_{km}$, $Q_{km-1}$ and $P_{km-2}$ is zero ensure that $x_{1}$, $y_{1}\not=0$, and hence that $C_{4}$, $C_{5}\not=0$. Clearly, for $j\geq N_{2}$,

[TABLE]

Note that, by the remarks following (2.27), all of these constants depend only on $s$. Note also that the condition $|\lambda_{1}|>1$ implies $\lim_{j\to\infty}|Q_{jkm-1}|=\lim_{j\to\infty}|Q_{jkm-2}|=\infty$.

∎

Lemma 9**.**

There exist positive constants $C_{8}<1$, $C_{9}$, $C_{10}$, $C_{11}$, $C_{12}$, $C_{13}$ and $C_{14}$ and positive integers $N_{5}$ and $N_{6}$, each depending only on $s$, such that, if $j\geq 1$, then

[TABLE]

if $j\geq N_{5}$, then

[TABLE]

and if $j\geq N_{6}$ and $n=jkm-1$ or $jkm-2$, then

[TABLE]

Proof.

From (2.48) it can be seen that $G(q)$ converges to $x_{1}$ (and thus, from (2.10) and (2.26), that $|G(q)|$ depends only on $s$) so that

[TABLE]

Set $C_{8}=|\lambda_{2}/\lambda_{1}|<1$, $C_{9}=|x_{1}-y_{1}|/(1+C_{8})$ and $C_{10}=|x_{1}-y_{1}|/(1-C_{8})$, and (2.49) follows. Note that $x_{1}\not=y_{1}$ (else the eigenvalues would be equal), so that $C_{9}$ and $C_{10}$ are non-zero.

Next, choose $N_{5}$ large enough so that $C_{8}^{N_{5}}<|y_{1}/x_{1}|$, and consider $j\geq N_{5}$. Thus,

[TABLE]

Set

[TABLE]

and (2.50) follows.

Finally, let $n=jkm-1$ or $jkm-2$, set $m^{{}^{\prime}}=\min\{C_{9},C_{11}\}$ and $M^{{}^{\prime}}=\max\{C_{10},C_{12}\}$. Choose $N_{6}$ such that $|G(q)|>M^{{}^{\prime}}C_{8}^{N_{6}}$. Let $j\geq N_{6}$. Then

[TABLE]

From the definitions of $m^{{}^{\prime}}$ and $M^{{}^{\prime}}$, and the choice of $N_{6}$, it follows that

[TABLE]

Set

[TABLE]

and (2.51) follows. ∎

Lemma 10**.**

Let $y$ be a another point on the unit circle. There exist positive constants $C_{15}$, $C_{16}$ and $C_{17}$ and positive integers $N_{9}$ and $N_{10}$, each depending only on $s$, such that if $j\geq N_{9}$, $n=jkm-1$ or $jkm-2$. and,

[TABLE]

then

[TABLE]

If $j\geq N_{10}$, $n=jkm-1$ or $jkm-2$ and the angle between $q$ and $y$ (measured from the origin) is less than $\pi/(2|\eta|)$, then

[TABLE]

and

[TABLE]

Proof.

Let

[TABLE]

Set $C_{15}^{\prime}=3+|G(q)|+C_{10}^{\prime}\,C_{8}^{N_{9}}$ and $C_{15}=2C_{15}^{\prime}/C_{2}^{\prime}$. Choose $N_{10}$ such that

[TABLE]

Let $j\geq N_{9}$. The inequalities at (2.45) and (2.46) imply that $|Q_{n}|\geq C_{2}^{\prime}C_{1}^{j}$. The condition on $N_{9}$ implies that, if $j\geq N_{9}$, then $|Q_{n}|-1/2>C_{2}^{\prime}\,C_{1}^{j}/2$. By similar reasoning to that used in the proof of (2.40), we find that

[TABLE]

Here we have used (2.49), (2.50), the bounds on $\epsilon_{1}$ and $\epsilon_{2}$ in the statement of the lemma and the inequality relating $|Q_{n}|$ and $C_{1}^{j}$ above.

Let $j\geq N_{10}$. As in the case where $\lambda_{1}=\lambda_{2}$,

[TABLE]

Here we have used (2.52) and once again the fact that the angle between $q$ and $y$, measured form the origin, is less than $\pi/(2|\eta|$) implies that $|y^{\eta}-q^{\eta}|\leq 2|\eta||q-y|$ (See the explanation before (2.43)). Using (2.49) and (2.50), it follows that,

[TABLE]

Set $C_{16}=\max\{4|\eta|/|G(q)|,\,\,4/|G(q)|\}$ and $C_{17}=4\,C_{15}/|G(q)|^{2}$. Statement (2.54) follows from (2.53) and (2.51). ∎

Lemma 11**.**

There exists an uncountable set of points on the unit circle such that, if $y$ is one of these points, then there exist two increasing sequences of integer, $\{n_{i}\}_{i=1}^{\infty}$ and $\{m_{i}\}_{i=1}^{\infty}$ say, such that

[TABLE]

for some $a$, $b\in\{1,2,\cdots,N_{G}\}$, where $a\not=b$.

Proof.

If $\lambda_{1}=\lambda_{2}$, we set $N^{\prime}=N_{8}$. If $|\lambda_{1}|>|\lambda_{2}|$, we set $N^{\prime}=N_{10}$. With the notation of Theorem 3, let $t\in S^{\diamond}$ and set $y=\exp(2\pi it)$. Let $c_{f_{n}}/d_{f_{n}}$ be one of the infinitely many approximants in the continued fraction expansion of $t$ satisfying (2.13) and (2.14), and set $x_{n}=\exp(2\pi ic_{f_{n}}/d_{f_{n}})$, so that $x_{n}$ is a primitive $d_{f_{n}}$-th root of unity and $H(x_{n})=H_{a}$. Let $\gamma(n)$ be as defined at (2.18). We use, in turn, the fact that chord length is shorter than arc length, a standard bound on the absolute value of the difference between a real number and an approximant in its continued fraction expansion, and (2.14), we find that

[TABLE]

Let $n^{\prime}=k\,N^{\prime}\,d_{f_{n}}^{2}-1$ or $k\,N^{\prime}\,d_{f_{n}}^{2}-2$. By (1), (2.17), (2.18), and (2.55) it follows that

[TABLE]

and similarly

[TABLE]

If $\lambda_{1}=\lambda_{2}$, then by (2.42), with $\epsilon=1/d_{f_{n}}^{2}$ and $j=N^{\prime}\,d_{f_{n}}$ (so that $j\geq N_{8}$), we find that

[TABLE]

If $|\lambda_{1}|\not=|\lambda_{2}|$ then (2.54) similarly implies that

[TABLE]

Thus, in either case,

[TABLE]

Similarly,

[TABLE]

The set $S^{\diamond}$ is uncountable because the conditions for membership require restrictions on only infinitely many of the partial quotients. One can easily construct a subset for which there is no restriction on a fixed infinite set of partial quotient. For each set of choices of positive integers for these partial quotients, one can choose other partial quotients so that the conditions for membership of $S^{\diamond}$ are fulfilled. Since the collection of all such continued fractions is uncountable, $S^{\diamond}$ is an uncountable set. Thus $Y_{G}=\{\exp(2\pi it):t\in S^{\diamond}\}$ is an uncountable set. ∎

Before proving Theorem 3, we show that $Y_{G}$ has measure zero. We use the following lemma.

Lemma 12**.**

$($[10]$\text{ pp. 140--141})$ ** Let $f(m)>1$ for $m=1,2,\dots$, and suppose that $\sum_{n=1}^{\infty}$ $1/f(m)$ $<\infty$. Then the set $S^{*}=\{t\in(0,1):e_{m}(t)>f(m)\text{ infinitely often}\,\,\}$ has measure zero.

Let $f(m)=2\pi\gamma(k\,N^{\prime}\,m^{2})$, where $\gamma(n)$ is the function at (2.18). Since $\gamma(j)\geq j$ for $j\geq 1$, it follows that $f(n)\geq 2\pi n^{2}$ and thus that $\sum_{n=1}^{\infty}1/f(n)$ converges. Since, for the regular continued fraction expansion of any real number, $d_{i}>i$ for $i\geq 4$, it follows that $d_{i}^{2}\geq(i+1)^{2}$ for $i\geq 4$, and thus it is clear from (2.14) that the elements in $S^{\diamond}$ satisfy $e_{m}(t)>f(m)$ infinitely often. Hence $S^{\diamond}$ (and thus $Y_{G}$) is a set of measure zero.

Of course the actual set of points on the unit circle at which $G(q)$ does not converge generally might have measure larger than zero.

Proof of Theorem 3. Let $y$ be any point in $Y_{G}$, where $Y_{G}$ is defined in the proof of Lemma 11, and let $t$ be the irrational in $(0,1)$ for which $y=\exp(2\pi it)$. $N^{\prime}$ is defined in Lemma 11. If $\lambda_{1}=\lambda_{2}$, we set $N_{1/2}=N_{1}$. If $|\lambda_{1}|>|\lambda_{2}|$, we set $N_{1/2}=N_{2}$.

Suppose $H(y)$ converges generally to $f\in\hat{\mathbb{C}}$ and that $\{v_{n}\}$, $\{w_{n}\}$ are two sequences such that

[TABLE]

Suppose first that $|g|<\infty$. By construction, there exist two infinite strictly increasing sequences of positive integers $\{n_{i}\}_{i=1}^{\infty}$, $\{m_{i}\}_{i=1}^{\infty}$ $\subset\mathbb{N}$ such that

[TABLE]

and

[TABLE]

for some $a\neq b$, where $a,b\in\{1,2,\cdots,N_{G}\}$. Also by construction each $n_{i}$ has the form $k\,N^{\prime}\,d_{k_{i}}^{2}-1$, where $d_{k_{i}}$ is some denominator convergent in the continued fraction expansion of $t$. A similar situation holds for each $m_{i}$. It can be further assumed that $L_{a}\neq g$, since $L_{a}\neq L_{b}$. For ease of notation write

[TABLE]

Write $P_{n_{i}}=Q_{n_{i}}(L_{a}+\epsilon_{n_{i}})$ and $P_{n_{i}-1}=Q_{n_{i}-1}(L_{a}+\delta_{n_{i}})$, where $\epsilon_{n_{i}}\to 0$ and $\delta_{n_{i}}\to 0$ as $i\to\infty$, so that

[TABLE]

where $\gamma_{n_{i}}\to 0$ as $i\to\infty$. Thus

[TABLE]

Because of (2.36) or (2.47), the fact that each $n_{i}$ has the form $k\,N^{\prime}\,d_{k_{i}}^{2}-1$, where $d_{k_{i}}$ is some denominator convergent in the continued fraction expansion of $t$ and (2.57), it follows that $Q_{n_{i}}/Q_{n_{i}-1}$ is absolutely bounded for $N^{\prime}\,d_{k_{i}}>N_{1/2}$. Therefore the right hand side of the last equality tends to 0 as $i\to\infty$ and thus

[TABLE]

Note that $|w_{n_{i}}|<\infty$ for all $i$ sufficiently large, since $|Q_{n_{i}}/Q_{n_{i}-1}|<\infty$. Similarly,

[TABLE]

By the (2.62), (2.63) and the triangle inequality,

[TABLE]

Thus

[TABLE]

Therefore $H(y)$ does not converge generally. The proof in the case where $g$ is infinite is similar.

Since $Y_{G}$ is uncountable, this proves the theorem.

$\Box$

Remark: Clearly $H(y)=y^{\eta}/G(y)$ converges generally if and only if $G(y)$ converges generally.

We have the following corollary to Theorem 3.

Corollary 2**.**

For each of the continued fractions $K(q)$, $S_{1}(q)$, $S_{2}(q)$ and $S_{3}(q)$, there exists an uncountable set of points on the unit circle at which the continued fraction does not converge generally.

Proof.

We use information contained in Table 1. In each case, $q$ $=$ $\exp$ $(2\pi ir/m)$, a primitive $m$-th root of unity and $m\equiv s\mod{d}$, where $(s,d)$ is the pair of integers from (2.8). $k$ is the integer and $f_{1},\cdots,f_{k}$ are the polynomials from the definition of the continued fraction $G(q)$ at (2.2). $H(q):=q^{\eta}/G(q)$, where $\eta$ is the rational in row one of the table.

Row three gives the value of $H(q)$, when $q$ $=$ $\exp$ $(2\pi ir/m)$ as above. $a_{km}$ is the $km$-th partial numerator in $G(q)$, as defined at (2.2).

The values in the first, third and last four rows come from the papers of Schur ([11]) and Zhang ([13]). The values of $a_{km}$ can be determined from the continued fractions at (1.1) and (1.4) – (1.6). For the last two entries in the $S_{1}(q)$ column, $\epsilon=(-1)^{(m-1)/4}$, this notation being employed to make the table fit the width of the page.

We give the proof for $S_{1}(q)$ only, since the proof for each of the other continued fractions is almost identical. One can easily check that $S_{1}(q)$ has the form given at (2.2) and satisfies the condition at (2.3), with $k=2$. From the table (or the paper of Zhang [13]), $S_{1}(q)$ satisfies (2.8) with $d=8$ and $s=1$. Likewise, (2.9) is satisfied with $\eta=1/8$. Conditions (2.10) are satisfied with $K_{0}=2$, $K_{1}=2$, $K_{2}=1$ and $K_{3}=K_{4}=1$ (when $m\equiv 1\mod{8}$). It is clear from row three of the table that (2.11) is satisfied, provided we choose $r\not\equiv u\mod{8}$. The conditions required by the theorem are satisfied, and the result follows. ∎

3. Concluding Remarks

In proving the existence of an uncountable set of points on the unit circle at which a $q$-continued fraction $G(q)$ does not converge in the general sense, our methods rely on knowing the behavior of the continued fraction at roots of unity and, if $q$ is a primitive $m$-th root of unity, on the fact that the values of $a_{km}(q)$, $P_{km-1}(q)$, $Q_{km-2}(q)$ and $Q_{km-1}(q)P_{km-2}(q)$ are fixed for $m$ belonging to certain arithmetic progressions (See (2.10)). Also important is the number $\eta$ from (2.9) which leads to the continued fraction $H(q)$ taking only finitely many values at roots of unity. Such $q$-continued fractions appear to be quite special and it would interesting to have a complete classification of them.

Our methods permit us to show the existence of a set of measure 0 at which each of the continued fractions diverges generally. We conjecture that each of these continued fraction diverges generally almost everywhere on the unit circle although at present we do not see how to prove this. It would be very interesting if a point on the unit circle which is not a root of unity could be exhibited at which any one of the continued fractions which are subject of Corollary 2 converged, in either the classical or general sense.

The most famous $q$-continued fraction after the Rogers-Ramanujan continued fraction is the Göllnitz-Gordon continued fraction, $GG(q)$ (see (2.4)). This continued fraction tends to the same limit as $S_{2}(q)$, for each $q$ inside the unit circle, but the behaviour at roots of unity is slightly different. As far as we are aware, its behaviour at roots of unity has not been studied. Based on computer investigations, it would seem that $GG(q)$ satisfies the conditions of Theorem 3 and thus that the Göllnitz-Gordon continued fraction diverges at uncountably many points on the unit circle. We hope to show this in a later paper.