You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: '[ We give another proof of a result of Adamczewski and Bell [@AuB16] concerning Mahler equations: A formal power series satisfying a $p$- and a $q$-Mahler equation over $\CX(x)$ with multiplicatively independent positive integers $p$ and $q$ is a rational function. The proof presented here is self-contained and essentially a compilation of proofs contained in a recent preprint [@MSRS16] of the authors. ]{}'
author:
- 'Reinhard Schäfke[^1], Michael F. Singer[^2]'
bibliography:
- 'DDrefs.bib'
date: 'March 21, 2017'
title: Mahler equations and rationality
---

We consider two Mahler operators, [[*i.e.*]{}]{} two endomorphisms $\sigma_j$, $j=1,2$, on the field $K=\CX[[x]][x^{-1}]$ of formal Laurent series with complex coefficients defined by $\sigma_1(f(x))=f(x^{p})$, $\sigma_2(f(x))=f(x^{q})$ for any $f(x)\in K$ where $p$ and $q$ are positive integers. Observe that $\sigma_1$ and $\sigma_2$ commute, [[*i.e.*]{}]{} $\sigma_1\sigma_2=\sigma_2\sigma_1$. We consider the field $\CX(x)$ of rational functions with complex coefficients as a subfield of $K$, the inclusion given by the expansion in a Laurent series at the origin. We want to prove the following theorem

Assume that $p$ and $q$ are multiplicatively independent, [[*i.e.*]{}]{} there are no nonzero integers $n_j$ such that $q^{n_2}=p^{n_1}$. Suppose that the formal series $f(x)\in K$ satisfies a system of two Mahler equations $$\label{sys2s}S_j(f(x))  =  \sigma_j^{m_j}(f(x)) + b_{j,m_j-1}(x) 
\sigma_j^{m_j-1}(f(x)) + \ldots + b_{j,0}(x)f(x) =  0,\ j=1,2$$ with $b_{j,i}(x) \in \CX(x)$.

Then $f(x)$ is rational.

[**Remark:**]{} 1. This Theorem was recently proved by Adamczewski and Bell in [@AuB16]. Their tools include a local-global principle to reduce the problem to a similar problem over finite fields, Chebotarev’s Density Theorem, Cobham’s Theorem and some asymptotics - all very different from the techniques used in the present work.\
2. [@AuB16] also provides background information about Mahler equations, in particular historical, many references to the literature and explains the relation to Cobham’s theorem in the theory of finite state machines. The fact that the generating functions of $p$-regular (and thus of $p$-automatic sequences) satisfy $p$-Mahler equations is shown in [@Becker94].\
3. The subsequent proof is essentially a compilation of work contained in our recent preprint [@MSRS16], see Corollary 15, part 3, and Proposition 19. The preprint presents a unified reduction theory of consistent pairs of first order systems of linear differential, difference, $q$-difference or Mahler equations like the one of Proposition \[2mprop\] below and uses it to deduce numerous statements on common solutions of two scalar linear differential, difference, $q$-difference or Mahler equations.

We may assume without loss of generality that $b_{j,0}(x)\neq0$, $j=1,2$. This follows from

\[linind\] Consider $w_1(x),...,w_\ell(x)\in K$ and a positive integer $m$. Then these series are $\CX(x)$-linearly dependent if and only if $w_1(x^m),...,w_\ell(x^m)$ are.

We only prove the nontrivial implication. Suppose that $w_1(x^m),...,w_\ell(x^m)$ are $\CX(x)$-linearly dependent, which means that there exist $a_k\in\CX(x)$, $k=1,...,\ell$, not all zero, such that $$a_1(x)w_1(x^{m})+...+a_\ell(x)w_\ell(x^{m})=0.$$ Now we can uniquely write $a_k(x)=\sum_{j=0}^{m-1} x^j c_k^{j}(x^{m})$, $k=1,...,\ell$, with rational functions $c_k^j(x)$. Expanding the terms in the above equation in Laurent series we obtain the equations $$c_1^j(x^{m})w_1(x^{m})+...+c_\ell^j(x^{m})w_\ell(x^{m})=0, j=0,...,m-1,$$ and hence $$c_1^j(x)w_1(x)+...+c_\ell^j(x)w_\ell(x)=0, j=0,...,m-1.$$ Since at least one of them must be nontrivial we obtain the linear dependence of the $w_j$, $j=1,...,\ell$. 

Consider now the $\CX(x)$-subspace $W$ of $K$ generated by $\sigma_1^m\sigma_2^r(f)$, $m=0,...,m_1-1,\ r=0,...m_2-1$. By (\[sys2s\]), $W$ is invariant under $\sigma_1$ and $\sigma_2$; here the fact that the $\sigma_j$ commute is used.

Let $g_1,...,g_n$ be [a]{} $\CX(x)$-basis of $W$ with $g_1=f$ and let $g=(g_1,...,g_n)^T$. Then we have that $$\label{sysW} \sigma_1(g)=A(x)g,\ \sigma_2(g)=B(x)g,$$ with $A,B\in \gl_n(\CX(x))$. By Lemma \[linind\], we actually have $A,B\in \GL_n(\CX(x))$ because the [components]{} of $\sigma_j(g)$ form again a basis of $W$.

Additionally, the coefficient matrices of (\[sysW\]) satisfy a certain [*consistency condition*]{}. Indeed, we have $$0=\sigma_1(\sigma_2(g))-\sigma_2(\sigma_1(g))=(\sigma_1(B)A-\sigma_2(A)B)g$$ and as the [components]{} of $g$ form a basis we obtain $$\label{2mcons}A(x^q)B(x)=B(x^p)A(x).$$

Our statement then follows from

\[2mprop\]Consider a system $$\label{2msys}y(x^p)=A(x)y(x),\ \ \ y(x^q)=B(x)y(x)$$ with multiplicatively independent positive integers $p$ and $q$ and $A(x),B(x)\in\GL_n(\CX(x))$ satisfying the consistency condition (\[2mcons\]). Suppose that $g(x)\in (\CX[[x]][x^{-1}])^n$ is a formal vectorial solution. Then $g(x)\in \CX(x)^n$.

Observe that we must actually have $n=1$ in the proof of the Theorem because the components of $g(x)$ are $\CX(x)$-linearly independent.

The proof of Proposition \[2mprop\] proceeds in three steps. We first prove that $g(x)$ converges in a neighborhood of $0$. In the second step (the heart of the proof) we show that $g(x)$ can be extended analytically to a meromorphic function on $\CX$ with only finitely many poles. Finally we prove that $g(x)$ has polynomial growth as $|x| \rightarrow \infty$ and therefore must be in $\CX(x)^n$. We begin with the first step.

\[conv\] The series $g(x)$ is convergent in a neighborhood of $0$.

This is a special case of [@Bez94], Theorem 1-2, and could also be deduced from [@Ran92], section 4. For the convenience of the reader, we provide a short proof. To do that, we truncate $g(x)$ at a sufficiently high power of $x$ to obtain $h(x)\in (\CX[x][x^{-1}])^n$ and introduce $r(x)=h(x)-A(x)^{-1}h(x^p)$ and $\tilde g(x)=g(x)-h(x)$. Then we have $$\label{2m-gr}
\tilde g(x)=A(x)^{-1}\tilde g(x^p) - r(x).$$ We denote the valuation of $A(x)^{-1}$ at the origin by $s\in\ZX$ and introduce $\tilde A(x)=x^{-s}A(x)^{-1}$ which is holomorphic at the origin.

First choose $M\in\NX$ such that $pM+s>M$ and $h(x)$ such that $g(x)-h(x)$ has at least valuation $M$. Then by (\[2m-gr\]), $r(x)$ also has at least valuation $M$. Now consider $R>0$ such that $\tilde A(x)$ is holomorphic and bounded on $D(0,R)$. Then consider for positive $\rho<\min(R,1)$ the vector space $E_\rho$ of all series $F(x)=\sum_{m=M}^\infty F_mx^m$ such that $\sum_{m=M}^\infty|F_m|{\rho}^m$ converges and define the norm $|F(x)|_\rho$ as this sum. Then $E_\rho$ equipped with $|\ |_\rho$ is a Banach space and the existence of a unique solution of (\[2m-gr\]) in $E_\rho$ for sufficiently small $\rho>0$ follows from the Banach fixed-point theorem using that $|x^s F(x^p)|_\rho\leq \rho^{Mp+s-M}|F(x)|_\rho$ for $F(x)\in E_\rho$. Since any solution $y(x) \in x^M\CX[[x]]$ of $y(x) = A(x)^{-1} y(x^p)$ must be zero, we have that $\tilde{g}(x)$ coincides with the solution in $E_\rho$. This proves the convergence of $\tilde g(x)$ and hence of $g(x)$.

We now turn to the task of showing that $g(x)$ can be extended to a meromorphic function on $\CX$. By (\[2msys\]), rewritten $g(x)=A(x)^{-1}g(x^p)$, the function $g$ can only be extended analytically to a meromorphic function on the unit disk. As we want to extend it [beyond]{} the unit disk, we use the change of variables $x=e^t$, $u(t)=y(e^t)$ and obtain a system of $q$-difference equations $$\label{2msys2q}u(pt)=\bar A(t)u(t),\ \ \ u(qt)=\bar B(t)u(t)$$ with $\bar A(t)=A(e^t)$, $\bar B(t)=B(e^t)$. It satisfies the consistency condition $$\label{2mconsi2q}\bar A(qt)\bar B(t)=\bar B(pt)\bar A(t).$$ Observe that $\bar A(t),\bar B(t)$ are not rational in $t$, but rational in $e^t$.

The heart of the proof of Proposition \[2mprop\] lies in understanding the behavior of solutions of (\[2msys2q\]). We do this by first showing in Lemma \[const\] that there is a formal gauge transformation $u=G\,v$, $G\in\GL_n(\CX\{t\}[t^{-1}])$, such that $v$ satisfies a system with constant coefficients. We then show in Lemma \[2mcont\] that the transformation matrix $G(t)$ and its inverse can be continued analytically to meromorphic functions on the $t$-plane. The “quotient” function $d(t) = G(t)^{-1}g(e^t)$ then satisfies a system with constant coefficients which can be solved explicitly. In this way, we show in Lemma \[d(t)\] that $d(t)$ can be extended analytically to an entire function on the Riemann surface of $\log(t)$. Using these three lemmas, we show in Lemma \[meromg\] that $g(x)$ can be continued analytically to a meromorphic function on the $x$-plane.

\[const\] There exists a convergent gauge transformation $u=G(t)\,v$, $G(t)\in\GL_n(\CX\{t\}[t^{-1}])$, such that $v$ satisfies $$\label{consteq}v(pt) = A_1 v(t) , \ \ \  v(qt) = B_1v(t)$$ where $A_1, B_1 \in \GL_n(\CX)$ commute.

Concerning the behavior at $t=0$, it is known that there exists a formal gauge transformation $u=G\,z$, $G\in\GL_n(\CX[[t^{1/s}]][t^{-1/s}])$, [$s\in\NX^*$,]{} that reduces $u(pt)=\bar A(t)u(t)$ to a system $z(pt)=t^DA_1z(t)$, where $D$ is a diagonal matrix with entries in $\frac1s\ZX$ and $A_1\in\GL_n(\CX)$ such that any eigenvalue $\lambda$ of $A_1$ satisfies $1\leq|\lambda|<|p|^{1/s}$, moreover $D$ and $A_1$ commute. If we write $D={ {\rm diag} }(d_1I_1,...,d_rI_r)$ with distinct $d_j$ and $I_j$ identity matrices of an appropriate size, then $A_1={ {\rm diag} }(A_1^{1},...,A_1^r)$ with diagonal blocks $A_1^{j}$ of corresponding size. $D$ and $A_1$ are essentially unique, [[*i.e.*]{}]{} except for a permutation of the diagonal blocks and passage from some $A_1^j$ to a conjugate matrix. If $D$ happens to be 0, then $s$ can be chosen to be 1 and $G$ is convergent (see [@PuSi], ch. 12, [@Ad31], [@Car12]).

Now by the consistency condition (\[2mconsi2q\]), the gauge transformation $v=B(t)u$ transforms $u(pt)=\bar A(t)u(t)$ to $v(pt)=\bar A(qt)v(t)$. The gauge transformation $v=G(qt)w$ then transforms this system to $w(pt)=(qt)^DA_1w(t)$. Now $(qt)^DA_1=t^{D}\,q^{D}A_1$ and there is a diagonal matrix $F$ with entries in $\frac1s\ZX$ commuting with $D$ and $A_1$ such that the gauge transformation $ w=t^{F}\tilde w$ reduces the latter system to $\tilde w(pt)=t^D\tilde A_1\tilde w(t)$, where $\tilde A_1=p^{-F}q^DA_1$ has again eigenvalues with modulus in $[1,|p|^{1/s}[$. Now we write $\tilde A_1={ {\rm diag} }(\tilde A_1^{1},...,\tilde A_1^r)$ and fix some $j\in\{1,...,r\}$. If $a^{j}_1,...,a^{j}_{r_j}$ are the eigenvalues of $A_1^j$ then $p^{-f_j}q^{d_j}a^j_\ell$, $\ell=1,...,r_j$, are those of $\tilde A_1^j$. By the uniqueness of the reduced form, the mapping $t\mapsto p^{-f_j}q^{d_j}t$ induces a permutation of the eigenvalues of $A_1^j$. If we apply it several times, if necessary, we obtain the existence of some $\ell\in\{1,...,r_j\}$ and of some positive integer $k$ such that $p^{-kf_j}q^{kd_j}a^j_\ell=a_\ell^j$. Due to our condition on $p$ and $q$ this is only possible if $d_j=0$. Thus we have proved that $D=0$ and $t=0$ is a so-called [*regular singular point*]{} of $u(pt)=\bar A(t)u(t)$.

We [therefore]{} obtain a matrix $A_1$ with eigenvalues $\lambda$ in the annulus $1\leq|\lambda|<p$ and $G(t)\in\GL_n(\CX\{t\}[t^{-1}])$ such that $u=G(t)v$ reduces the first equation of (\[2msys2q\]) to $v(pt)=A_1v(t)$. This means $$\label{2mg1}G(pt)=\bar A(t)G(t)A_1^{-1}\mbox{ for small }t.$$ Applying the same gauge transformation to the second equation of (\[2msys2q\]) yields an equation $v(qt)=\tilde{\bar B}(t)v(t)$ with some $\tilde{\bar B}(t)\in\GL_n(C\{t\}[t^{-1}])$. It satisfies the consistency condition $A_1\tilde{\bar B}(t)=\tilde{\bar B}(pt)A_1$. Now we expand $\tilde{\bar B}(t)=\sum_{m=m_0}^\infty C_m t^m$. The coefficients satisfy $A_1C_m=C_m (p^mA_1)$, $m\geq m_0$. As $A_1$ and $p^mA_1$ have no common eigenvalue unless $m=0$, we obtain that $\tilde{\bar B}(t)=:B_1$ is constant and commutes with $A_1$. We note the second equation satisfied by $G$ $$\label{2mg1b}G(qt)=\bar B(t)G(t)B_1^{-1}\mbox{ for small }t.$$

\[2mcont\] The functions $G(t)^{\pm1}$ can be continued analytically to meromorphic functions on $\CX$ and there exists $\delta>0$ such that both can be continued analytically to the sectors $\{t\in\CX^*\mid \delta<\arg(\pm t)<2\delta\}$.

Let $\calM$ be the set of poles of $\bar A(t)^{\pm1}$, [[*i.e.*]{}]{} the set of $t$ such that $e^t$ is a pole of $\bar A(x)$ or $\bar A(x)^{-1}$. Note that $\calM$ is $2 \pi i$-periodic, has no finite accumulation point and is contained in some vertical strip $\{t\in\CX\mid -D<{\operatorname{Re}}t<D\}$. By (\[2mg1\]), $G(t)^{\pm1}$ can be continued analytically to $\CX^*\setminus(\calM\cdot p^\NX)$ and thus to meromorphic functions on $\CX$ which we denote by the same name. By construction, $G(t)^{\pm1}$ are also analytic in some punctured neighborhood of the origin. By the properties of $\calM$, the infimum of the $|{\operatorname{Re}}t_1| $ on the set of all $t_1\in\calM$ having nonzero real part is a positive number. As $\calM$ is contained in some vertical strip there exist sectors $\{t\in\CX^*\mid \delta<\arg(\pm t)<2\delta\}$ disjoint to $\calM$ and hence to $\calM\cdot p^\NX$. Therefore $G(t)^{\pm1}$ can be analytically continued to these sectors and the lemma is proved.

\[d(t)\] The function $d(t)=G(t)^{-1}g(e^t)$ can be continued analytically to the Riemann surface of $\log(t)$.

By Lemma \[2mcont\] and because $g(x)$ is holomorphic in some punctured neighborhood of $x=0$ by Lemma \[conv\], $d(t)$ is defined and holomorphic for some sector $S=\{t\in\CX\mid |t|>K,\ \pi+\delta<\arg t<\pi+2\delta\}$. By (\[2msys\]), (\[2mg1\]), and (\[2mg1b\]) it satisfies $$\label{2md}d(pt)=A_1d(t),\ \ d(qt)=B_1d(t)\mbox{ for }t\in S.$$ To solve (\[2md\]), consider a matrix $L_1$ commuting with $B_1$ such that $p^{L_1}=A_1$. Put $F(t)=t^{-L_1}d(t)$. Then $$\label{2mf}F(pt)=F(t),\ \ F(qt)=\tilde B_1F(t)\mbox{ for }t\in S$$ where [$\tilde B_1=B_1q^{-L_1}$]{}. Thus $H(s)=F(e^s)$ is $\log(p)$-periodic on the half-strip $B=\{s\in\CX\mid {\operatorname{Re}}s>\log(K),\ \pi+\delta<{\mbox{\rm Im\,}}s<\pi+2\delta\}$ and can be expanded in a Fourier series. This implies that $$\label{2mfourier}F(t)=\sum_{\ell=-\infty}^\infty F_\ell\,t^{\frac{2 \pi i}{\log(p)}\ell}\mbox{ for }t\in S.$$ The second equation of (\[2mf\]) yields conditions on the Fourier coefficients $$F_\ell\exp\left(2\pi i\tfrac{\log(q)}{\log(p)}\ell\right)=\tilde B_1F_\ell\mbox{ for }\ell\in\ZX.$$ Therefore $F_\ell=0$ unless $\exp\left(2\pi i\frac{\log(q)}{\log(p)}\ell\right)$ is an eigenvalue of $\tilde B_1$. Since $p$ and $q$ are multiplicatively independent, the quotient $\frac{\log(q)}{\log(p)}$ is irrational and hence $\exp\left(2\pi i\frac{\log(q)}{\log(p)}\right)$ is not a root of unity. Therefore all the numbers $\exp\left(2\pi i\frac{\log(q)}{\log(p)}\ell\right)$, $\ell\in\ZX$ are different and only finitely many of them can be eigenvalues of $\tilde B_1$. This shows that the Fourier series (\[2mfourier\]) has finitely many terms and thus $F(t)$ can be analytically continued to the whole Riemann surface $\hat \CX$ of $\log(t)$. The same holds for $d(t)=t^{L_1}F(t)$.

\[meromg\] The function $g(x)$ can be continued analytically to a meromorphic function on $\CX$ with finitely many poles.

[**Remark:**]{} According to Theorem 4.2 of [@Ran92] (see also [@BCR13]), it is sufficient to show that $g(x)$ [does not have]{} the unit circle as a natural boundary and the rationality of $g(x)$ follows. [We show how it follows naturally, in our context,]{} that $g(x)$ can be continued analytically as a meromorphic function to all of $\CX$ and, as well, that it has only finitely many poles. The rationality of $g(x)$ then follows as in [@Ran92] and [@BCR13] from a growth estimate (Lemma \[grow\]).

The function $h(t)=g(e^t)$ is holomorphic for $t$ with large negative real part by Lemma \[conv\] and $2\pi i$-periodic. Using Lemma \[2mcont\] we conclude that $h(t)=G(t)d(t)$ can be analytically continued to a meromorphic function on $\hat \CX$, in particular the point $t=2\pi i$ is at most a pole of $h$. By its periodicity, this implies that $t=0$ also is at most a pole of $h$ and that it can be continued analytically to a meromorphic function on $\CX$ which we denote by the same name.

Since $h(t)=g(e^t)$ for $t$ with large negative real part, $h(t)$ is $2\pi i$-periodic for those values of $t$, hence also its analytic continuation to a meromorphic function on all of $\CX$. This periodicity allows [one]{} to define a meromorphic function $\tilde g(x)$ on [$\CX\backslash\{0\}$]{} by $\tilde g(e^t)=h(t)$. As $\tilde g(x)=g(x)$ for small $|x|\neq0$ by the construction of $h$, we have shown that $g(x)$ can be continued analytically to a meromorphic function on $\CX$ which will again be denoted by the same name.

The formula $h(t)=G(t)d(t)$ and Lemma \[2mcont\] also imply that $h$ is analytic in the sector $\tilde S=\{t\in\CX^*\mid \delta<\arg t<2\delta\}$. As this sector contains some half strip $\{t\in\CX\mid {\operatorname{Re}}t>L,\mu{\operatorname{Re}}t<{\mbox{\rm Im\,}}t<\mu{\operatorname{Re}}t+3\pi\}$ for some positive $L,\mu$ which has vertical width larger than $2\pi$ and $h$ is $2\pi i$-periodic, its poles are contained in some vertical strip $\{t\in\CX\mid -L<{\operatorname{Re}}t<L\}$. This implies that $g(x)$ has only a finite number of poles.

The proof of Proposition \[2mprop\] is completed once we have shown

\[grow\]The function $g(x)$ has polynomial growth as $|x|\to\infty$.

This is shown in the proof of Theorem 4.2 in [@Ran92] (see also [@BCR13]). For the convenience of the reader, we reproduce it below.

Consider $r_0{>1}$ such that $g(x)$ and $A(x)$ are holomorphic on the annulus $|x|>r_0/2$. There are positive numbers $K,M$ such that $|A(x)|\leq K|x|^M$ for $|x|\geq r_0$. Consider now the annuli $$\calA_j=\{x\in\CX\mid r_0^{p^j}\leq|x|<r_0^{p^{j+1}}\},\ j=0,1,...$$ covering the annulus $|x|\geq r_0$. Any $x\in\calA_j$ can be written $x=\xi^{p^j}$ with some $\xi\in\calA_0$. Then we estimate using (\[2msys\]) and the inequality for $|A(x)|$ $$|g(x)|=|g(\xi^{p^j})|\leq K ^j \left(|\xi|^{p^{j-1}}\cdots|\xi|^p|\xi|\right)^M \max_{r_0\leq|\xi|\leq r_0^p}|g(\xi)|.$$ Hence there is a positive constant $L$ such that $|g(x)|\leq L\, K^j \,|x|^{\frac M{p-1}}\mbox{ for }x\in\calA_j.$ Assuming $\log(r_0)\geq1$ without loss in generality, we find that $j\leq \log(\log(|x|))/\log(p)$ for $x\in\calA_j$. Hence there exists $d>0$ such that $$|g(x)|\leq L\, (\log(|x|))^d\,|x|^{\frac M{p-1}}\mbox{ for }|x|>r_0.$$

[ [Acknowledgement.]{} The authors would like to thank Boris Adamczewski for suggesting an improvement of the proof of Lemma \[meromg\] and for pointing out the article [@BCR13] to us.]{}

[^1]: Institut de Recherche Mathématique Avancée, Université de Strasbourg et C.N.R.S., 7, rue René Descartes, 67084 Strasbourg Cedex, France, [<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>]{}.

[^2]: Department of Mathematics, North Carolina State University, Box 8205, Raleigh, NC 27695, USA, [<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>]{}.