The Convergence and Divergence of $q$-Continued Fractions outside the Unit Circle

TL;DR

This paper investigates the convergence properties of two classes of $q$-continued fractions outside the unit circle, providing theorems that establish conditions for their convergence and that of their odd and even parts.

Contribution

It introduces new convergence theorems for specific $q$-continued fractions outside the unit circle, expanding understanding of their limit behavior.

Findings

Convergence of $q$-continued fractions outside the unit circle is established.

Conditions for the convergence of odd and even parts are provided.

Theorems guarantee convergence at points outside the unit circle.

Abstract

We consider two classes of $q$-continued fraction whose odd and even parts are limit 1-periodic for $|q|>1$, and give theorems which guarantee the convergence of the continued fraction, or of its odd- and even parts, at points outside the unit circle.

Equations

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

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

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

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

$$1-\frac{x}{1}\text{lower}5.83333pt+\frac{x^2}{1}\text{lower}4.30554pt-\frac{x^3}{1}\text{lower}5.83333pt+\text{lower}5.83333pt\cdots$$

$$\frac{x}{1}\text{lower}5.83333pt+\frac{x^4}{1}\text{lower}5.83333pt+\frac{x^8}{1}\text{lower}5.83333pt+\frac{x^{12}}{1}\text{lower}5.83333pt+\text{lower}5.83333pt\cdots .$$

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

$$b_0+\frac{b_2{a}_1}{b_2{b}_1+a_2}\text{lower}4.30554pt-\frac{a_2{a}_3{b}_4/b_2}{a_4+b_3{b}_4+a_3{b}_4/b_2}\text{lower}4.30554pt-\frac{a_4{a}_5{b}_6/b_4}{a_6+b_5{b}_6+a_5{b}_6/b_4}\text{lower}4.30554pt-\text{lower}5.83333pt\cdots .$$

$$\begin{gathered} \frac{b_0{b}_1+a_1}{b_1}-\frac{a_1{a}_2{b}_3/b_1}{b_1(a_3+b_2{b}_3)+a_2{b}_3}\text{lower}4.30554pt-\frac{a_3{a}_4{b}_5{b}_1/b_3}{a_5+b_4{b}_5+a_4{b}_5/b_3} \\ \text{lower}4.30554pt-\frac{a_5{a}_6{b}_7/b_5}{a_7+b_6{b}_7+a_6{b}_7/b_5}\text{lower}4.30554pt-\frac{a_7{a}_8{b}_9/b_7}{a_9+b_8{b}_9+a_8{b}_9/b_7}\text{lower}4.30554pt-\text{lower}5.83333pt\cdots . \end{gathered}$$

$$(i)\text{parabolic,}$$

$$(ii)\text{elliptic,}$$

$$(iii)\text{loxodromic,}$$

$$G(q):$$

$$\text{lower}5.83333pt+\frac{f_1(q^1)}{1}\text{lower}5.83333pt+\text{lower}5.83333pt\cdots \text{lower}5.83333pt+\frac{f_k(q^1)}{1}\text{lower}5.83333pt+\text{lower}5.83333pt\cdots \text{lower}5.83333pt+\frac{f_1(q^n)}{1}\text{lower}5.83333pt+\text{lower}5.83333pt\cdots \text{lower}5.83333pt+\frac{f_k(q^n)}{1}\text{lower}5.83333pt+\text{lower}5.83333pt\cdots ,$$

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

$$\text{degree}(a_{i+1}(q))=\text{degree}(a_i(q))+m,$$

$$G_e(q):=1+\frac{a_1}{1+a_2}\text{lower}4.30554pt-\frac{a_2{a}_3}{a_4+a_3+1}\text{lower}4.30554pt-\frac{a_4{a}_5}{a_6+a_5+1}\text{lower}4.30554pt-\text{lower}5.83333pt\cdots$$

$$\approx 1+\frac{\frac{a_1}{1+a_2}}{1}\text{lower}4.30554pt-\frac{\frac{a_2{a}_3}{(1+a_2)(a_4+a_3+1)}}{1}\text{lower}4.30554pt-\frac{\frac{a_4{a}_5}{(a_4+a_3+1)(a_6+a_5+1)}}{1}\text{lower}4.30554pt-\text{lower}5.83333pt\cdots$$

$$=1+K_{n=1}^{\infty }\frac{c_n}{1},$$

$$c_n$$

$$\underset{n\to \infty }{\lim }{c}_n$$

$$=\frac{1}{(1+q^m)(1+q^{-m})}:=c.$$

$$G_o(q):=\frac{1+a_1}{1}-\frac{a_1{a}_2}{a_3+a_2+1}\text{lower}4.30554pt-\frac{a_3{a}_4}{a_5+a_4+1}\text{lower}4.30554pt-\frac{a_5{a}_6}{a_7+a_6+1}\text{lower}4.30554pt-\text{lower}5.83333pt\cdots .$$

$$\begin{gathered} G(q)=1+\frac{6q}{1}\text{lower}5.83333pt+\frac{3q^2+7q}{1}\text{lower}5.83333pt+\frac{3q^3+5q^2}{1}\text{lower}5.83333pt+\frac{q^4+7q^3+3q+2}{1}\text{lower}5.83333pt+ \\ \frac{q^5+3q^4+2q^3}{1}\text{lower}5.83333pt+\frac{q^6+2q^5+7q^3}{1}\text{lower}5.83333pt+\frac{q^7+7q^5}{1}\text{lower}5.83333pt+\frac{q^8+7q^6+3q^3+2q}{1}\text{lower}5.83333pt+\text{lower}5.83333pt\cdots \\ \text{lower}5.83333pt\cdots \text{lower}5.83333pt+\frac{q^{4n+1}+3q^{3n+1}+2q^{2n+1}}{1}\text{lower}5.83333pt+\frac{q^{4n+2}+2q^{3n+2}+7q^{2n+1}}{1} \\ \text{lower}5.83333pt+\frac{q^{4n+3}+5q^{3n+2}+2q^{2n+3}}{1}\text{lower}5.83333pt+\frac{q^{4n+4}+7q^{3n+3}+3q^{2n+1}+2q^n}{1}\text{lower}5.83333pt+\text{lower}5.83333pt\cdots \end{gathered}$$

$$f_1(x)$$

$$f_2(x)$$

$$f_3(x)$$

$$f_4(x)$$

$$a_{4n+j}(q)=f_j(q^n).$$

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

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

$$+$$

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

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

$$\text{degree}(a_{i+1}(q))$$

$$\text{degree}(b_i(q))$$

$$\text{degree}(a_n(q))=(n-1)a+r_1,$$

Full text

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

1. Introduction

Studying the convergence behaviour of the odd and even parts of continued fractions is interesting for a number of different reasons (see, for example, Section 9.4 of [6]). In this present paper, we examine the convergence behaviour of $q$-continued fractions outside the unit circle.

Many well-known $q$-continued fractions have the property that their odd and even parts converge everywhere outside the unit circle. These include the Rogers-Ramanujan continued fraction,

[TABLE]

and the three Ramanujan-Selberg continued fractions studied by Zhang in [8], namely,

[TABLE]

[TABLE]

and

[TABLE]

It was proved in [1] that if $0<|x|<1$ then the odd approximants of $1/K(1/x)$ tend to

[TABLE]

while the even approximants tend to

[TABLE]

This result was first stated, without proof, by Ramanujan. In [8], Zhang expressed the odd and even parts of each of $S_{1}(q)$, $S_{2}(q)$ and $S_{3}(q)$ as infinite products, for $q$ outside the unit circle.

Other $q$-continued fractions have the property that they converge everywhere outside the unit circle. The most famous example of this latter type is Göllnitz-Gordon continued fraction,

[TABLE]

In this present paper we study the convergence behaviour outside the unit circle of two families of $q$-continued fractions, families which include all of the above continued fractions.

2. Convergence of the odd and even parts of $q$-continued fractions

outside the unit circle

Before coming to our theorems, we need some notation and some results on limit 1-periodic continued fractions.

Let the $n$-th approximant of the continued fraction $b_{0}+K_{n=1}^{\infty}a_{n}/b_{n}$ be $P_{n}/Q_{n}$. The even part of $b_{0}+K_{n=1}^{\infty}a_{n}/b_{n}$ is the continued fraction whose $n$-th numerator (denominator) convergent equals $P_{2n}$ ($Q_{2n}$), for $n\geq 0$. The odd part of $b_{0}+K_{n=1}^{\infty}a_{n}/b_{n}$ is the continued fraction whose zero-th numerator convergent is $P_{1}/Q_{1}$, whose zero-th denominator convergent is $1$, and whose $n$-th numerator (respectively denominator) convergent equals $P_{2n+1}$ (respectively $Q_{2n+1}$), for $n\geq 1$.

For later use we give explicit expressions for the odd- and even parts of a continued fraction. From [7], page 83, the even part of $b_{0}+K_{n=1}^{\infty}a_{n}/b_{n}$ is given by

[TABLE]

From [7], page 85, the odd part of $b_{0}+K_{n=1}^{\infty}a_{n}/b_{n}$ is given by

[TABLE]

Definition:. Let $t(w)=c/(1+w)$, where $c\not=0$. Let $x$ and $y$ denote the fixed points of the linear fractional transformation $t(w)$. Then $t(w)$ is called

[TABLE]

In case (iii), if $|1+x|>|1+y|$, then $\lim_{n\to\infty}t^{n}(w)=x$ for all $w\not=y$, $x$ is called the attractive fixed point of $t(w)$ and $y$ is called the repulsive fixed point of $t(w)$.

Remark: The above definitions are usually given for more general linear fractional transformations but we do not need this full generality here.

The fixed points of $t(w)=c/(1+w)$ are $x=(-1+\sqrt{1+4c})/2$ and $y=(-1+\sqrt{1+4c})/2$. It is easy to see that $t(w)$ is parabolic only in the case $c=-1/4$, that it is elliptic only when $c$ is a real number in the interval $(-\infty,-1/4)$ and that it is loxodromic for all other values of $c$.

Let $\hat{\mathbb{C}}$ denote the extended complex plane. From [7], pp. 150–151, one has the following theorem.

Theorem 1**.**

Suppose $1+K_{n=1}^{\infty}a_{n}/1$ is limit $1$-periodic, with $\lim_{n\to\infty}a_{n}=c\not=0$. If $t(w)=c/(1+w)$ is loxodromic, then $1+K_{n=1}^{\infty}a_{n}/1$ converges to a value $f\in\hat{\mathbb{C}}$.

Remark: In the cases where $t(w)$ is parabolic or elliptic, whether $1+K_{n=1}^{\infty}a_{n}/1$ converges or diverges depends on how the $a_{n}$ converge to c.

We also make use of Worpitzky’s Theorem (see [7], pp. 35–36).

Theorem 2**.**

(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 $|w|\leq 1/2$.

We first consider continued fractions of the form

[TABLE]

where $f_{s}(x)\in\mathbb{Z}[q][x]$, 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 and the three Ramanujan-Selberg continued fractions are of this form, with $k$ at most 2. Following the example of these four continued fractions, we make the additional assumptions that, for $i\geq 1$,

[TABLE]

where $m$ is a fixed positive integer, and that all of the polynomials $a_{n}(q)$ have the same leading coefficient. We prove the following theorem.

Theorem 3**.**

Suppose $G(q)=1+K_{n=1}^{\infty}a_{n}(q)/1$ is such that the $a_{n}:=a_{n}(q)$ satisfy (2.4) and (2.5). Suppose further that each $a_{n}(q)$ has the same leading coefficient. If $|q|>1$ then the odd and even parts of $G(q)$ both converge.

Remark: Worpitzky’s Theorem gives only that odd- and even parts of $G(q)$ converge for those $q$ satisfying $|(1+q^{m})(1+q^{-m})|>4$, a clearly weaker result.

Proof.

Let $|q|>1$. For ease of notation we write $a_{n}$ for $a_{n}(q)$. By (2.1), the even part of $G(q)$ is given by

[TABLE]

where, for $n\geq 3$,

[TABLE]

By (2.5), the fact that each of the $a_{i}(q)$’s has the same leading coefficient and the fact that if $|q|>1$ then $\lim_{i\to\infty}1/a_{i}=0$, it follows that

[TABLE]

Hence $G_{e}(q)$ is limit $1$-periodic. Note that the value of $c$ depends on $q$.

Let the fixed points of $t(w)=c/(1+w)$ be denoted $x$ and $y$. From the remarks following (2.3), it is clear that $t(w)$ is parabolic only in the case $-1/((1+q^{m})(1+q^{-m}))=-1/4$. The only solution to this equation is $q^{m}=1$, so that $t(w)$ is not parabolic for any point outside the unit circle.

Similarly, $t(w)$ is elliptic only when $-1/((1+q^{m})(1+q^{-m})=-1/4-v$, for some real positive number $v$. The solutions to this equation satisfy $q^{m}=(i+\sqrt{v})/(i-\sqrt{v})$ or $q^{m}=(i-\sqrt{v})/(i+\sqrt{v})$. However, it is easily seen that these are points on the unit circle.

In all other cases $t(w)$ is loxodromic and $G_{e}(q)$ converges in $\hat{\mathbb{C}}$. This proves the result for $G_{e}(q)$.

Similarly, by (2.2), the odd part of $G(q)$ is given by

[TABLE]

The proof in this case is virtually identical. ∎

As an application of the above theorem, we have the following example.

Example 1**.**

If $|q|>1$, then the odd and even parts of

[TABLE]

converge.

Proof.

Let $k=4$ and

[TABLE]

Then, for $n\geq 0$ and $1\leq j\leq 4$,

[TABLE]

Thus (2.4) is satisfied. It is clear that (2.5) is satisfied with $M=1$ and each $a_{n}(q)$ has the same leading coefficient, namely, 1. ∎

Remark: It is clear form Theorem 3 that if $k=1$ and $f_{i}(x)$ is any polynomial with coefficients in $\mathbb{Z}[q]$, then the odd and even parts of $1+K_{n=0}^{\infty}f_{1}(q^{n})/1$ converge everywhere outside the unit circle to values in $\hat{\mathbb{C}}$, since all the conditions of the theorem are satisfied automatically, at least for a tail of the continued fraction.

We also 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]

An example of a continued fraction of this type is the Göllnitz-Gordon continued fraction (with $k=1$).

We suppose that degree $(a_{1}(q))=r_{1}$, degree $(b_{0}(q))=r_{2}$, and that, for $i\geq 1$,

[TABLE]

where $a$ and $b$ are fixed positive integers and $r_{1}$ and $r_{2}$ are non-negative integers. Condition 2.7 means that, for $n\geq 1$,

[TABLE]

We also supposed that each $a_{n}(q)$ has the same leading coefficient $L_{a}$ and that each $b_{n}(q)$ has the same leading coefficient $L_{b}$.

For such continued fractions we have the following theorem.

Theorem 4**.**

Suppose $G(q)=b_{o}+K_{n=1}^{\infty}a_{n}(q)/b_{n}(q)$ is such that the $a_{n}:=a_{n}(q)$ and the $b_{n}:=b_{n}(q)$ satisfy (2.6) and (2.7). Suppose further that each $a_{n}(q)$ has the same leading coefficient $L_{a}$ and that each $b_{n}(q)$ has the same leading coefficient $L_{b}$. If $2b>a$ then $G(q)$ converges everywhere outside the unit circle. If $2b=a$, then $G(q)$ converges outside the unit circle to values in $\hat{\mathbb{C}}$, except possibly at points $q$ satisfying $L_{b}^{2}/L_{a}q^{b-r_{1}+2r_{2}}\in\left[-4,0\right)$. If $2b<a$, then the odd and even parts of $G(q)$ converge everywhere outside the unit circle.

Proof.

. Let $|q|>1$. We first consider the case $2b>a$. By a simple transformation, we have that

[TABLE]

Since $2b>a$, $a_{n}/(b_{n}b_{n-1})\to 0$ as $n\to\infty$, and $G(q)$ converges to a value in $\hat{\mathbb{C}}$, by Worpitzky’s theorem.

Suppose $2b=a$. Then, by (2.7), (2.8) and the fact that each $a_{n}(q)$ has the same leading coefficient $L_{a}$ and that each $b_{n}(q)$ has the same leading coefficient $L_{b}$,

[TABLE]

Note once again that the value of $c$ depends on $q$. Once again, by the remarks following (2.3), the linear fractional transformation $t(w)=c/(1+w)$ is parabolic only in the case $L_{a}/(L_{b}^{2}q^{b-{r_{1}}+2\,{r_{2}}})=-1/4$ or $q^{b-r_{1}+2r_{2}}=-4L_{a}/L_{b}^{2}$.

Similarly, $t(w)$ is elliptic only when $q^{-b+{r_{1}}-2\,{r_{2}}}\,{L_{a}}/{{L_{b}}}^{2}\in\left(-\infty,-1/4\right)$, or

[TABLE]

for some real positive number $v$. In other words, $t(w)$ is elliptic (for $|q|>1$) only when $q^{b-r_{1}+2r_{2}}$ lies either in the open interval $(-4L_{a}/L_{b}^{2},0)$ or $(0,-4L_{a}/L_{b}^{2})$, depending on the sign of $L_{a}$. In all other cases, $t(w)$ is loxodromic, and $G(q)$ converges.

Suppose $2\,b<a$. From (2.1) it is clear that the even part of $G(q)=b_{0}+K_{n=1}^{\infty}a_{n}/b_{n}$ can be transformed into the form $b_{0}+K_{n=1}^{\infty}c_{n}/1$, where, for $n\geq 3$,

[TABLE]

Once again using (2.7), (2.8) and the fact that each $a_{n}(q)$ has the same leading coefficient $L_{a}$ and that each $b_{n}(q)$ has the same leading coefficient $L_{b}$, we have that

[TABLE]

The linear fractional transformation $t(w)=c/(1+w)$ is parabolic only in the case $-q^{2b-a}/(1+q^{2b-a})^{2}=-1/4$ or $q^{2b-a}=1$, and thus $|q|=1$. It is elliptic only when $-q^{2b-a}/(1+q^{2b-a})^{2}\in(-\infty,-1/4)$, and a simple argument shows that this implies that $|q^{2b-a}|=1$, and again $|q|=1$.

In all other cases $t(w)$ is loxodromic, and the even part of $G(q)$ converges by Theorem 1.

The proof for the odd part of $G(q)$ is very similar and is omitted. ∎

Remarks: (1) Worpitzky’s Theorem once again gives weaker results. In the example below, for example, Worpitzky’s Theorem gives that $G(q)$ converges for $|q|>4$, in contrast to the result from our theorem, which says that $G(q)$ converges everywhere outside the unit circle, except possibly for $q\in[-4,-1)$.

(2) In some cases the result is the best possible. Numerical evidence suggests that the continued fraction below converges nowhere in the interval $(-4,-1)$.

As an application of Theorem 4, we have the following example.

Example 2**.**

If $|q|>1$, then

[TABLE]

converges, except possibly for $q\in[-4,-1)$.

Proof.

Let $k=4$ and

[TABLE]

Then, for $n\geq 0$ and $1\leq j\leq 4$,

[TABLE]

The other requirements of the theorem are satisfied, with $L_{a}=L_{b}=1$, $a=2$, $b=1$, $r_{1}=2$ and $r_{2}=1$. Therefore $b-r_{1}+2r_{2}=1$, $L_{a}/L_{b}^{2}=1$ and $G(q)$ converges outside the unit circle, except possibly for $q\in[-4,-1)$. ∎