1 Introduction

The genesis of q-calculus goes back to Fermat and his computation of \(\int _0^a t^ \alpha dt\), \( \alpha >0\), by dividing the interval [0, a] using a geometric dissection of ratio \(0<q<1\), see [2, Ch. 10]. This led to the development of Jackson’s q-integral, which is the inverse of Jackson q-derivative

$$\begin{aligned} d_qf(x):=\frac{f(qx)-f(x)}{qx-x}. \end{aligned}$$
(1)

In the current literature, it is common to work with the dilation operator

$$\begin{aligned} \sigma _q(f)(x):=f(qx), \end{aligned}$$

instead. Note that both are related by \(\sigma _q=(q-1)x d_q+\text {id}\), where \(\text {id}\) is the identity.

Other sources of q-analogues are Euler’s works on partitions, leading to q-exponentials, Gauss’ q-binomial formula and Heine’s q-hypergeometric series with applications in Number Theory, see [18]. These are examples of functions y satisfying q-difference equations, i.e., relations of the form

$$\begin{aligned} H(x;y,\sigma _qy,\dots ,\sigma _q^ny)=0,\qquad q\in {\mathbb {C}}^*. \end{aligned}$$

Writing this equation in terms of \(d_q\) and noticing that \(d_qf\rightarrow f'\) as \(q\rightarrow 1\), we find that q-difference equations are a discrete counterpart to differential equations. For historical accounts of q-calculus and q-difference equations we refer to the survey [15].

In the analytic setting, linear algebraic q-difference equations have been studied since the works of Carmichael [9], Birkhoff [5] and Adams [1]. Despite an initial lagged development in comparison with ordinary differential equations (ODEs), Birkhoff’s program on the subject has been successfully carried out. This includes the study of symmetries, analytic classification, normal forms, and the inverse Riemann problem. We can mention the works of J.-P. Bézivin [3], L. Di Vizio [13], J. P. Ramis [24], J. Sauloy and C. Zhang [26], and the references therein.

As it is usual in analytic problems, divergent power series solutions emerge at irregular singular points. In the q-difference framework when \(|q|>1\), the divergence is given by the factor \(|q|^{sn^2/2}, s>0\). These series are referred as q-Gevrey. Optimal values for s are usually found using Newton polygon techniques [8, 22], which can be extended to more intricate equations including derivatives [29]. In fact, this is the first step in the study of summability of these formal solutions and their Stokes phenomena. Several approaches for q-summability have been developed, taking into account different notions of asymptotic expansions on usual sectors or q-spirals. They adapt the use Borel and Laplace transformations by suitable q-analogues. For instance, using the Jacobi’s theta function [25, 30], the two q-analogues to the exponential as kernels for the Laplace transform, both with respect to Riemann’s and Jackson’s integrals [14, 16, 27, 28].

Returning to the differential case, after the systematization of summability and its applications in the study of ODEs, the theory was also applied in the setting of singularly perturbed problems, see [6]. However, for problems such as doubly-singular systems of analytic ODEs of the form

$$\begin{aligned} \epsilon ^ \alpha x^{p+1} \frac{\partial y}{\partial x}=F(x,\epsilon ,y), \end{aligned}$$
(2)

new ideas to identify the correct source of divergence of power series solutions were necessary. In fact, (2) led to the development of monomial summability in [7]. The key here is to recognize the monomial variable \(t=\epsilon ^ \alpha x^p\) as the correct one to compute asymptotic expansions. In (2) we have that \(y=(y_1,\ldots ,y_N)\in {\mathbb {C}}^N\) is a vector of unknown functions, \(p, \alpha \in {\mathbb {N}}^+\), F is analytic at \((0,0,0)\in {\mathbb {C}}\times {\mathbb {C}}\times {\mathbb {C}}^N\), with \(F(0,0,0)=0\), and \(DF_y(0,0,0)\) is invertible. The singular perturbation in \(\epsilon \) occurs since the nature of the equation changes from differential to implicit as \(\epsilon \rightarrow 0\). We point out that some q-analogues of singularly perturbed ODEs has been studied in [19, 21] and their references, but only with expansions in the perturbation parameter.

It is precisely (2) the inspiration of this work. Our goal is to describe the divergence of the power series solutions to singularly perturbed q-difference equations obtained by discretizing (2). Our aim is aligned with the understanding of “the complete theory of convergence and divergence of formal series” [4, p. 222] for these systems. More specifically, for \(q\in {\mathbb {C}}\) with \(|q|>1\), we consider the problems

$$\begin{aligned} \epsilon ^{\alpha }x^{p+1}d_{q,x}(y)(x,\epsilon )=F(x,\epsilon ,y),\qquad \epsilon ^{\alpha }x^{p}\sigma _{q,x}(y)(x,\epsilon )=F(x,\epsilon ,y), \end{aligned}$$

with similar hypotheses as before. Here we use the notation \(\sigma _{q,x}(y)(x,\epsilon ):=y(qx,\epsilon )\) and \((q-1)x d_{q,x}+\text {id}=\sigma _{q,x}\) to indicate the action on the first coordinate. However, when the context is clear we will omit this index.

Although similar, we decided to analyze each one of them by separate, specially because the system involving \(d_{q,x}\) is better suited for confluence and it makes sense for \(p=-1\). As we mentioned before, although it is common to work only with equations involving \(\sigma _q\), a direct approach using \(d_q\) can also be fruitful, see, e.g., [17, 27]. In fact, \(\sigma _q\) and \(d_q\) motivate different types of q-summability [14].

For each equation we will first establish the existence and uniqueness of a solution

$$\begin{aligned} {\hat{y}}(x,\epsilon )=\sum _{n=0}^\infty y_n(\epsilon )x^n=\sum _{n=0}^\infty u_n(x)\epsilon ^n\in {\mathbb {C}}[[x,\epsilon ]]^N, \end{aligned}$$
(3)

such that \({\hat{y}}(0,0)=0\). Then we will determine the growth of the families \(\{y_n(\epsilon )\}_{n\in {\mathbb {N}}}\) and \(\{u_n(x)\}_{n\in {\mathbb {N}}}\) by using majorant series and adequate families of norms, including a new adaptation of Nagumo norms [23], that we call q-Nagumo norms. Incidentally, we can also treat by the same technique both equations in the Fuchsian-like case, namely, when \(p=0\). More precisely, we have the following theorem, see below for notation.

Theorem 1

Fix \(q\in {\mathbb {C}}\) such that \(|q|>1\). Consider each one of the systems

$$\begin{aligned} \epsilon ^ \alpha x^{p+1}d_{q,x}(y)(x,\epsilon )&=F(x,\epsilon ,y), \end{aligned}$$
(4)
$$\begin{aligned} \epsilon ^ \alpha x^{p}\sigma _{q,x}(y)(x,\epsilon )&=F(x,\epsilon ,y), \end{aligned}$$
(5)

where \(y\in {\mathbb {C}}^N\), \(p\in {\mathbb {N}}\), \( \alpha \in {\mathbb {N}}^+\), F is analytic at \((0,0,0)\in {\mathbb {C}}\times {\mathbb {C}}\times {\mathbb {C}}^N\), \(F(0,0,0)=0\), and \(DF_y(0,0,0)\) is an invertible matrix. Then, each system has a unique formal power series solution \({\hat{y}}(x,\epsilon )\in {\mathbb {C}}[[x,\epsilon ]]^N\) of the form (3) such that \({\hat{y}}(0,0)=0\). Moreover, we have the following results.

  1. 1.

    If \(p>0\), there is \(r>0\) such that \(y_n\in {\mathcal {O}}_b(D_r)\), \(u_n\in {\mathcal {O}}_b(D_{r/|q|^{\lfloor \frac{n}{ \alpha }\rfloor }})\), and there are constants \(C=C(q),A=A(q)>0\) such that, for all \(n\ge 0\),

    $$\begin{aligned} \sup _{|\epsilon |\le r} |y_n(\epsilon )|\le CA^n |q|^{\frac{n^2}{2p}},\qquad \sup _{|x|\le {r}/{|q|^{\lfloor \frac{n}{ \alpha }\rfloor }} } |u_n(x)|\le C A^n. \end{aligned}$$
  2. 2.

    If \(p=0\), there is \(r>0\) such that \(y_n\in {\mathcal {O}}_b(D_{r/|q|^{n/ \alpha }})\), \(u_n\in {\mathcal {O}}_b(D_{r/|q|^{\lfloor n/ \alpha \rfloor }})\), and there are constants \(C=C(q),A=A(q)>0\) such that, for all \(n\ge 0\),

    $$\begin{aligned} \sup _{|\epsilon |\le r/|q|^{\frac{n}{ \alpha }}} |y_n(\epsilon )|\le CA^n,\qquad \sup _{|x|\le {r}/{|q|^{\lfloor \frac{n}{ \alpha }\rfloor }} } |u_n(x)|\le C A^n. \end{aligned}$$

The divergence of the solutions manifests in two ways. In some cases the growth of the coefficients is given by a power of \(|q|^{n^2/2}\), which is expected for q-difference equations. But it is also evidenced in the reduction of the radii of the disks employed. For \(u_n(x)\) the appearance of the radius \(r/|q|^{\lfloor \frac{n}{ \alpha }\rfloor }\) means that, when solved recursively, we need to reduce the radius by a factor of q every \( \alpha \) steps.

The same proof can be extended to (4) for \(p=-1\). We have obtained the following result, with higher rate of divergence, in comparison with Theorem 1.

Theorem 2

Fix \(q\in {\mathbb {C}}\) such that \(|q|>1\). Consider the system

$$\begin{aligned} \epsilon ^ \alpha d_{q,x}(y)(x,\epsilon )=F(x,\epsilon ,y), \end{aligned}$$
(6)

with the same hypothesis as in Theorem 1. Then, (6) has a unique formal power series solution \({\hat{y}}(x,\epsilon )=\sum _{n=0}^\infty u_n(x)\epsilon ^n\), where \({\hat{y}}(0,0)=0\) and \(u_n\in {\mathcal {O}}_b(D_{r/|q|^{\lfloor \frac{n}{ \alpha } \rfloor }})\), for some \(r>0\). Moreover, there are constants \(C=C(q),A=A(q)>0\) such that, for all \(n\ge 0\),

$$\begin{aligned} \sup _{|x|\le {r}/{|q|^{\lfloor \frac{n}{ \alpha } \rfloor }} } |u_n(x)|\le C A^n |q|^{n^2/2 \alpha ^2}. \end{aligned}$$

These are interesting problems and our results leave an open door to the problem of adapting monomial summability to this context. Several problems are already evident, such as the lose of symmetry in the growth of the coefficients in each variable. In the differential case, when \(p= \alpha =1\), both \(y_n\) and \(u_n\) are defined in commons disks, for all n, and both grow as n!. This is no longer valid in the q-difference case and new strategies will be necessary to understand how to associate analytic solutions asymptotic to the formal ones obtained here. This is also the reason why we do not expand the solutions in the corresponding monomial or why we do not lift the main equation as in [12] (technique useful for certain families of holomorphic PDEs). These questions will be addressed in a future work.

The plan for the paper is as follows. In Sect. 2 we recall the basics on some q-analogues, \(d_q\) and \(\sigma _q\), while Sect. 3 includes some spaces of q-Gevrey series. In Sect. 4 we develop q-Nagumo norms and their properties. Section 5 is devoted to prove Theorems 1 and 2. Then, we include examples in Sect. 6, showing that the bounds provided by our results are optimal. We also include an appendix with non-modified q-Nagumo norms (\(q>1\)) better suited for confluence. In fact, they help recovering the divergence rate of formal solutions of (2) using confluence.

Notation. \({\mathbb {N}}\) is the set of non-negative integers and \({\mathbb {N}}^+:={\mathbb {N}}\setminus \{0\}\). We work in \(({\mathbb {C}}^2,{ 0})\) with local coordinates \((x_1,x_2)\), \({\mathbb {C}}[[x_1,x_2]]\) and \({\mathbb {C}}\{x_1,x_2\}\) will denote the spaces of formal and convergent power series in \((x_1,x_2)\) with coefficients in \({\mathbb {C}}\). We will also write \((x_1,x_2)=(x,\epsilon )\), to distinguish the singular parameter \(\epsilon \). Given \(R>0\), we will write \(D_{R}:=\{x\in {\mathbb {C}}: |x|<R\}\). We set \({\mathcal {O}}(\Omega )\) (resp. \({\mathcal {O}}_b(\Omega )\)) for the space of \({\mathbb {C}}\)-valued holomorphic (resp. and bounded) functions on an open domain \(\Omega \subseteq {\mathbb {C}}\). Note that \({\mathcal {O}}_b(\Omega )\) endowed with the supremum norm is a Banach space.

2 Preliminaries

We start by recalling some q-analogues, including q-factorials and q-Gevrey series. From now on we fix \(q\in {\mathbb {C}}\) with \(|q|>1\). We write

$$\begin{aligned} [\lambda ]_q=\frac{q^\lambda -1}{q-1},\qquad \lambda \in {\mathbb {R}}, \lambda \ge 0, \end{aligned}$$

for the q-analogue to \(\lambda \), where the principal branch of the logarithm is considered for \(\lambda >0\), and which reduces to

$$\begin{aligned} [n]_q=1+q+\cdots +q^{n-1},\qquad \text {for } n\in {\mathbb {N}}^+. \end{aligned}$$

We have that \([0]_q=0\) and

$$\begin{aligned} [np]_q=\frac{q^{p}-1}{q-1} \frac{q^{np}-1}{q^p-1}= [p]_q\cdot [n]_{q^p},\qquad n,p\in {\mathbb {N}}^+. \end{aligned}$$
(7)

For future use, we note that \(|[n]_q|\le [n]_{|q|}\), for \(n\in {\mathbb {N}}\), and

$$\begin{aligned} \lim _{n\rightarrow +\infty } \frac{[n]_q}{q^n}=\frac{1}{q-1}. \end{aligned}$$
(8)

These constants appear naturally while considering Jackson’s q-derivative (1) of a function f. As before, \(\sigma _q(f)(x):=f(qx)\). For analytic functions \(f\in {\mathcal {O}}(D_r)\),

$$\begin{aligned} \sigma _q(f), d_q(f)\in {\mathcal {O}}(D_{r/|q|}). \end{aligned}$$
(9)

They can be computed term by term using power series expansion and the rules

$$\begin{aligned} d_q(x^n)=[n]_q x^{n-1},\qquad \sigma _q(x^n)=q^n x^n,\qquad n\in {\mathbb {N}}. \end{aligned}$$

These formulas allow to consider \(d_q, \sigma _d:{\mathbb {C}}[[x]]\rightarrow {\mathbb {C}}[[x]]\), also defined term by term. In this setting, Leibniz rule is replaced by

$$\begin{aligned} d_q(fg)(x)=d_q(f)(x)g(x)+f(qx)d_q(g)(x). \end{aligned}$$
(10)

We recall the coefficients

$$\begin{aligned} (a;q)_n=\prod _{j=0}^{n-1} (1-aq^j),\qquad (a;q^{-1})_\infty =\prod _{j=0}^{\infty } (1-aq^{-j}),\qquad a\in {\mathbb {C}}. \end{aligned}$$
(11)

The second one is convergent as we can compare it with a geometric series. The q-factorial is defined accordingly as

$$\begin{aligned} [n]_q^{!}=[1]_q [2]_q\cdots [n]_q=\frac{(q;q)_n}{(1-q)^n}. \end{aligned}$$

In general, for \(|q|>1\), since \(\lambda \in {\mathbb {R}}\longmapsto [\lambda ]_{|q|}\) is a strictly increasing function, the same holds for the map \(n\in {\mathbb {N}}\longmapsto [n]^{!}_{|q|}\). Therefore,

$$\begin{aligned} & \frac{[n-p]_{|q|}}{([n]^{!}_{|q|})^{1/p}} \\ & \quad = \frac{[n-p]_{|q|}}{([n]_{|q|}\cdots [n-p+1]_{|q|})^{1/p} ([n-p]^{!}_{|q|})^{1/p}}&\le \frac{[n-p]_{|q|}}{[n-p+1]_{|q|} } \frac{1}{([n-p]^{!}_{|q|})^{1/p}}. \end{aligned}$$

Thus

$$\begin{aligned} \frac{[n-p]_{|q|}}{([n]^{!}_{|q|})^{1/p}} \le \frac{1}{([n-p]^{!}_{|q|})^{1/p}},\qquad \text {for integers } n>p>0.\end{aligned}$$
(12)

Another useful q-analogue is the q-binomial coefficient

$$\begin{aligned} \left[ \begin{array}{l}{n}\\ {j}\end{array}\right] _{q} = \frac{[n]^{!}_{q}}{[j]^{!}_{q} [n-j]^{!}_{q}},\qquad 0\le j\le n, \end{aligned}$$

which is a polynomial in q of degree \(j(n-j)\) satisfying q-Pascal’s formula

$$\begin{aligned} \left[ \begin{array}{l}{n}\\ {j}\end{array}\right] _{q}=\left[ \begin{array}{l}{n-1}\\ {j-1}\end{array}\right] _{q}+q^j \left[ \begin{array}{l}{n-1}\\ {j}\end{array}\right] _{q}, \end{aligned}$$
(13)

see, e.g., [18, Chapter 6]. In particular, it follows that \(\left[ \begin{array}{l}{n}\\ {j}\end{array}\right] _{|q|} \ge 1\), i.e.,

$$\begin{aligned} [j]^{!}_{|q|} [n-j]^{!}_{|q|}\le [n]^{!}_{|q|}. \end{aligned}$$
(14)

which will be used later.

Remark

A sequence \(\{M_n\}_{n\ge 1}\) of positive real numbers with \(M_0=1\), satisfies \(M_nM_k\le M_{n+k}\), for all \(n,k\ge 0\), when it is log-convex, i.e., when it holds that

$$\begin{aligned} M_n^2\le M_{n-1} M_{n+1}, \end{aligned}$$

for all \(n\ge 1\). This fact follows easily using induction on k. An example is \(M_n=([n]_{|q|}^{!})^s\), \(s>0\), \(|q|>1\). Indeed, \(M_{n}/M_{n-1}=[n]_{|q|}^s\) which is increasing in n.

In general, q-factorials determine the divergence rate of solutions of singular q-difference equations. Two classical examples are the following.

Example 1

A q-analogue to Euler’s equation is

$$\begin{aligned} x^2d_qy(x)+y(x)=x \quad \text { having as a solution }\quad {\hat{E}}_q(x):=\sum _{n=0}^\infty (-1)^n [n]_q^{!} x^{n+1}, \end{aligned}$$

which is divergent for \(|q|>1\). In contrast, the q-difference equation

$$\begin{aligned} x\sigma _qy(x)=y(x)-x \quad \text { has the solution } \quad {\hat{Y}}_q(x)=\sum _{n=0}^\infty q^{n(n+1)/2} x^{n+1}, \end{aligned}$$

which also diverges for \(|q|>1\). As \(|q|\rightarrow 1^+\), the divergence of \({\hat{E}}(x)=\sum _{n=0}^\infty (-1)^nn! x^{n+1}=\lim _{|q|\rightarrow 1^+} {\hat{E}}_q(x)\) persists, while \(\lim _{|q|\rightarrow 1^+} {\hat{Y}}_q(x)=\frac{x}{1-x}\) converges.

In this framework, the following notion plays the role of Gevrey series.

Definition 1

A series \({\hat{f}}=\sum _{n\ge 0} a_n x^n\in {\mathbb {C}}[[x]]\) is s-q-Gevrey, where \(s\ge 0\) and \(|q|>1\), if there are constants \(C=C(q),A=A(q)>0\) such that

$$\begin{aligned} |a_n|\le CA^n |q|^{sn^2/2}. \end{aligned}$$
(15)

We will denote the space of such series by \({\mathbb {C}}[[x]]_{q,s}\).

Note that the q-factorial is desirable when working with q-difference equations with \(d_q\), while \(|q|^{n^2/2}\) is natural when the equation is written in terms of \(\sigma _q\).

Remark

We can interchange the term \(|q|^{n^2/2}\) above for \(|q|^{n(n\pm 1)/2}\). Also, we can use the terms \([n]_{|q|}^{!}\) or \(|[n|_q^{!}|\), i.e., asking for a series to be s-q-Gevrey is equivalent to request that

$$\begin{aligned} |a_n|\le D B^n ([n]_{|q|}^{!})^s,\qquad n\in {\mathbb {N}}, \end{aligned}$$

for some constants \(B,D>0\) that might also depend on q. Indeed, since

$$\begin{aligned} [n]_q=q^{n-1}[n]_{q^{-1}}, \quad \text { and }\quad [n]_{q}^{!}=q^{n(n-1)/2} [n]_{q^{-1}}^{!}, \end{aligned}$$

we have \([n]_{q}^{!}=(q^{-1};q^{-1})_{n}\frac{q^{n(n+1)/2}}{(q-1)^n}\), which entails the limit

$$\begin{aligned} \lim _{n\rightarrow +\infty } [n]_{q}^{!}/{\displaystyle \frac{q^{\frac{n(n+1)}{2}}}{(q-1)^n}}=\lim _{n\rightarrow +\infty }(q^{-1};q^{-1})_n=(q^{-1};q^{-1})_\infty =c(q). \end{aligned}$$

This means that \([n]_{q}^{!}\) is asymptotically equivalent to \( c(q)(q-1)^{-n}q^{n(n+1)/2}\), c.f., [24, p. 55]. Thus, \([n]_{|q|}^{!}\) is asymptotically equivalent to \( c(|q|)(|q|-1)^{-n} |q|^{n(n+1)/2}\), and \(|[n|_q^{!}|\) is to \( |c(q)| |q-1|^{-n} |q|^{n(n+1)/2}\). We also note the simple bound

$$\begin{aligned} \frac{c(|q|)}{(|q|-1)^{n}}|q|^{\frac{n(n+1)}{2}}\le [n]_{|q|}^{!}\le \frac{1}{(|q|-1)^n} |q|^{\frac{n(n+1)}{2}}, \end{aligned}$$

relating \(|q|^{\frac{n(n+1)}{2}}\) and \([n]_{|q|}^{!}\), up to geometric terms.

It is easy to check that \({\mathbb {C}}[[x]]_{q,s}\) is stable under sums, products and the operators \(d_q\) and \(\sigma _q\). Regarding ramifications and changes in s, we highlight that:

  1. 1.

    \({\hat{f}}\) is s-q-Gevrey if and only if it is 1-\(q^s\)-Gevrey.

  2. 2.

    Fix \(p\in {\mathbb {N}}^+\) and write \({\hat{f}}(x)=\sum _{j=0}^{p-1} x^j {\hat{f}}_j(x^p)\), where \({\hat{f}}_j(t)=\sum _{n=0}^\infty a_{np+j} t^n\). Then \({\hat{f}}(x)\) is s-q-Gevrey (in x) if and only if each \({\hat{f}}_j(t)\) is sp-\(q^p\)-Gevrey (in t). In fact, we obtain from (15) that

    $$\begin{aligned} |a_{np+j}|\le CA^{np+j} |q|^{s(np+j)^2/2} = C_j A_j^n |q|^{sp^2n^2/2}, \end{aligned}$$

    where \(C_j=CA^j|q|^{sj^2/2}\), \(A_j=A^p |q|^{pjs}\). The converse is similar.

3 Some spaces of q-Gevrey series

This section introduces two spaces of q-Gevrey series describing the phenomena encountered in Theorem 1.

Let \({\hat{f}}\in {\mathbb {C}}[[x_1,x_2]]\) be a formal power series, written canonically as

$$\begin{aligned} {\hat{f}}=\sum _{n,m=0}^\infty a_{n,m} x_1^n x_2^m = \sum _{n=0}^\infty y_n(x_2) x_1^n=\sum _{m=0}^\infty u_m(x_1) x_2^m, \end{aligned}$$
(16)

for \(y_n\in {\mathbb {C}}[[x_2]]\) and \(u_m\in {\mathbb {C}}[[x_1]]\). First, we consider the space

$$\begin{aligned} {\mathcal {O}}_{0}^q:=\left\{ {\hat{f}}\in {\mathbb {C}}[[x_1,x_2]]: |a_{n,m}|\le CA^{n+m} |q|^{nm}, \text { for some } C,A>0 \text { and all }n,m \right\} , \end{aligned}$$

that in some sense plays the roles of the ring \({\mathcal {O}}_0={\mathbb {C}}\{x_1,x_2\}\) of convergent power series, for the case \(q=1\). We can characterize their elements as follows.

Lemma 1

Let \({\hat{f}}\in {\mathbb {C}}[[x_1,x_2]]\) as in (16). The following are equivalent:

  1. 1.

    \({\hat{f}}\in {\mathcal {O}}_{0}^q\),

  2. 2.

    For some \(r,B,D>0\), \(y_n\in {\mathcal {O}}_b(D_{r/|q|^n})\), for all n, and

    $$\begin{aligned} \sup _{|x_2|\le r/|q|^n} |y_n(x_2)|\le DB^n,\end{aligned}$$
  3. 3.

    For some \(r,B,D>0\), \(u_m\in {\mathcal {O}}_b(D_{r/|q|^m})\), for all m, and

    $$\begin{aligned} \sup _{|x_1|\le r/|q|^m} |u_m(x_1)|\le DB^m.\end{aligned}$$

Proof

To show that (1) implies (2) and (3), fix \({\hat{f}}\in {\mathcal {O}}_{0}^q\). Then

$$\begin{aligned} |y_n(x_2)|\le \sum _{m=0}^\infty CA^{n+m} |q^nx_2|^{m}\le 2CA^n,\qquad \text { for } |x_2|\le 1/(2A|q|^n), \end{aligned}$$

and analogously for \(u_m(x_1)\). Now, if (2) holds, by Cauchy’s inequalities,

$$\begin{aligned} |a_{n,m}|=\left| \frac{1}{m!}\frac{\partial ^m y_n}{\partial x_2^m}(0)\right| \le \frac{DB^n}{(\rho /|q|^n)^m}=DB^n\rho ^{-m} |q|^{nm},\qquad \text { for } 0<\rho <r, \end{aligned}$$

and for all \(n,m\in {\mathbb {N}}\). Thus (1) is valid. In the same way, (3) implies (1). Alternatively, note the symmetry in the previous definition: \({\hat{f}}(x_1,x_2)\in {\mathcal {O}}_{0}^q\) if and only if \({\hat{f}}(x_2,x_1)\in {\mathcal {O}}_{0}^q\), so we can exchange the role of the variables. \(\square \)

Thus, Theorem 1 (2) claims that the formal solutions of (4), (5) (\(p=0\)) belong to \({\mathcal {O}}_0^{q^{1/ \alpha }}\).

Example 2

Consider the series

$$\begin{aligned} M(x_1,x_2):=\sum _{n,m=0}^\infty q^{nm} x_1^n x_2^m=\sum _{n=0}^\infty \frac{x_1^n}{1-q^nx_2}=\sum _{m=0}^\infty \frac{x_2^m}{1-q^m x_1}\in {\mathcal {O}}_0^q, \end{aligned}$$

confluent to \(\left[ (1-x_1)(1-x_2)\right] ^{-1}\) when \(q\rightarrow 1\). Although M does not define an analytic function at (0, 0), given any \(\delta >0\) the series converges on the set

$$\begin{aligned} U_{q,\delta }^1:=U_{q,\delta }\times D_{\delta },\qquad U_{q,\delta }:=\left\{ x_1\in {\mathbb {C}}:\, |1-q^mx_1|>\delta ^m\hbox { for all } m\ge 0\right\} . \end{aligned}$$

The second variable consists of a disc of radius \(\delta >0\). In the first variable we have the intersection of the complement of circles with radius \((\delta /|q|)^m\) and center \(1/q^m\). If \(\delta =1\), these circles have 0 at the boundary. If additionally q is real and positive, then \(U_{q,1}=\{x_1\in {\mathbb {C}}: |1-x_1|>1\}\) since \(D(1/q^{m+1},1/q^{m+1})\subset D(1/q^m,1/q^m)\). For general values of q and \(\delta \), these circles revolve around the origin, see Fig. 1. Since \(M(x_1,x_2)=M(x_2,x_1)\), M also defines an analytic function on \(U_{q,\delta }^2=D_\delta \times U_{q,\delta }\).

Finally, we note that \(M(x,\epsilon )\) is the unique formal power series solution of

$$\begin{aligned} \epsilon \sigma _{q,x}y=y-\frac{1}{1-x}, \end{aligned}$$

corresponding to equation (5) in Theorem 1(2).

Fig. 1
figure 1

Circles in \(U_{q,\frac{1}{2}}\), for \(q=\frac{3}{2}e^{i\pi /4}\)

Full size image

The second type of series that appear naturally in Theorem 1 are the following, named after the analogy with the differential case.

Definition 2

Fix a monomial \(x_1^px_2^ \alpha \). A series \({\hat{f}}=\sum a_{n,m} x_1^n x_2^m\in {\mathbb {C}}[[x_1,x_2]]\) is 1-q-Gevrey in \(x_1^px_2^ \alpha \) if there are constants \(C=C(q),A=A(q)>0\) such that

$$\begin{aligned} |a_{n,m}|\le CA^{n+m}\min \{ |q|^{n^2/2p}, |q|^{nm/ \alpha } \}. \end{aligned}$$

The space of these series will be denoted by \({\mathcal {O}}^q_{x_1^p x_2^ \alpha }\).

Proceeding as in Lemma 1, we can establish the following characterization.

Lemma 2

Let \({\hat{f}}\) be as in (16). The following assertions are equivalent:

  1. 1.

    \({\hat{f}}\in {\mathcal {O}}^q_{x_1^p x_2^ \alpha }\),

  2. 2.

    For some \(r,B,D>0\), \(y_n\in {\mathcal {O}}_b(D_{r})\), \(u_n\in {\mathcal {O}}_b(D_{r/|q|^{n/ \alpha }})\) for all n, and

    $$\begin{aligned} \sup _{|x_2|\le r} |y_n(x_2)|\le DB^n|q|^{n^2/2p},\qquad \sup _{|x_1|\le r/|q|^{n/ \alpha }} |u_n(x_1)|\le DB^n. \end{aligned}$$

We collect some properties of these spaces. The proof is straightforward.

Lemma 3

Consider \(|q|>1\) and \(x_1^p x_2^ \alpha \). The following assertions hold:

  1. 1.

    \({\mathcal {O}}^{q}_{0}\) and \({\mathcal {O}}^q_{x_1^p x_2^ \alpha }\) are rings stable by the operators \(d_{q,x_j}, \sigma _{q,x_j}\), \(j=1,2\)

  2. 2.

    \({\mathcal {O}}^q_{x_1^p x_2^ \alpha }\subset {\mathcal {O}}^{q^{1/ \alpha }}_{0}\).

  3. 3.

    \({\hat{f}}=\sum _{j=0}^{p-1} {\hat{f}}_j(x_1^p,x_2) x_1^j\in {\mathcal {O}}_{0}^q\) if and only if \({\hat{f}}_j(z,x_2)\in {\mathcal {O}}_{0}^{q^{p}}\), for \(j=0,1,\dots ,p-1\).

  4. 4.

    \({\hat{f}}=\sum _{0\le j<p, 0\le l< \alpha } {\hat{f}}_{j,l}(x_1^p,x_2^ \alpha ) x_1^j x_2^l\in {\mathcal {O}}^q_{x_1^px_2^ \alpha }\) if and only if \({\hat{f}}_j(z,\eta )\in {\mathcal {O}}_{z\eta }^{q^{p}}\), for all \(j=0,1,\dots ,p-1\), \(l=0,1,\dots , \alpha -1\).

We conclude this section remarking that the series \({\hat{y}}=\sum a_{n,m} x_1^n x_2^m\) appearing in Theorem 2 are precisely those satisfying bounds of the type

$$\begin{aligned} |a_{n,m}|\le CA^{n+m} |q|^{nm/ \alpha }\cdot |q|^{n^2/2 \alpha ^2}, \end{aligned}$$

for certain constants \(C=C(q), A=A(q)>0\). In contrast, this is a higher divergence rate compared to the one describes in the spaces \({\mathcal {O}}_o^q\) and \({\mathcal {O}}_{x_1^px_2^ \alpha }^q\).

4 The q-Nagumo norms

To treat the divergence generated by the singular parameter in equations (4), (5), and (6), we introduce an adaptation of modified Nagumo norms. These were introduced in [6] to study the divergence of solutions of singular perturbatated ODEs. In turn, the former were based on classical Nagumo norms, introduced in M. Nagumo in his work [23] on power series solutions of analytic PDEs.

Fix \(|q|>1\), \(0<\rho <r\), and \(n\in {\mathbb {N}}\). Consider \(d_n:D_{r/|q|^n}\rightarrow {\mathbb {R}}\) given by

$$\begin{aligned} d_n(t)={\left\{ \begin{array}{ll} r-|q^nt|, & \rho /|q|^n\le |t|\le r/|q|^n,\\ r-\rho , & |t|\le \rho /|q|^n. \end{array}\right. } \end{aligned}$$

Choosing \(\rho \) adequately, the following assertions hold, see Fig. 2.

Fig. 2
figure 2

The auxiliary functions \(d_n(t)\)

Full size image

Lemma 4

If \(\rho =r/|q|\), then

  1. 1.

    \(|d_n(t)-d_n(s)|\le |q|^n|t-s|\).

  2. 2.

    \(d_n(qt)=d_{n+1}(t)\), for \(|t|\le r/|q|^{n+1}\).

  3. 3.

    \(d_{n+1}(t)\le d_n(t)\le r(1-1/|q|)\), for all \(n\in {\mathbb {N}}\) and \(|t|\le r/|q|^{n+1}\).

From now on we fix the value \(\rho =r/|q|\) and consider \(d_n\) accordingly. For \(f\in {\mathcal {O}}_b(D_r)\), we define the nth q-Nagumo norm by

$$\begin{aligned} \Vert f\Vert _n:=\sup _{|x|\le r/|q|^n} |f(x)|\cdot d_n(|x|)^n. \end{aligned}$$

To simplify notation we omit the dependence on r and q. Note that for \(n=0\), \(\Vert f\Vert _0\) is simply the supremum norm. Also, if \(\Vert f\Vert _n\) is finite, we have that

$$\begin{aligned} |f(x)|\le \frac{\Vert f\Vert _n}{d_n(|x|)^n},\qquad \text { for } |x|<r/|q|^n. \end{aligned}$$

This family of norms takes into account that factor \(r-|q^nx|\) that measures correctly the distance of x to the boundary of the disk \(D_{r/|q|^n}\). In this way, we obtain information on the operators \(d_q\) and \(\sigma _q\), as we shall see it in the following lemma.

Lemma 5

Let \(|q|>1\), \(n,m\in {\mathbb {N}}\) and \(f,g \in {\mathcal {O}}_b(D_r)\). Then:

  1. 1.

    \(\Vert f+g\Vert _n\le \Vert f\Vert _n+\Vert g\Vert _n\) and \(\Vert fg\Vert _{n+m}\le \Vert f\Vert _n \Vert g\Vert _m\).

  2. 2.

    \(\Vert d_q(f)\Vert _{n+1}\le 2|q|^{n+1}\Vert f\Vert _n\).

  3. 3.

    \(\Vert \sigma _q(f)\Vert _{n+1}\le r(1-\frac{1}{|q|})\Vert f\Vert _n\).

Proof

For (1), using Lemma 4(3), we see that

$$\begin{aligned} |f(x)g(x)|d_{n+m}(|x|)^{n+m}\le |f(x)|d_n(|x|)^n\, |g(x)| d_m(|x|)^m\le \Vert f\Vert _n \Vert g\Vert _m, \end{aligned}$$

for all \(|x|\le r/|q|^{n+m}\). This proves the inequality for the product.

For (2) fix x such that \(|x|\le r/|q|^{n+1}\). To bound \(d_q(f)\), consider first the case \(|x|\ge \rho /|q|^{n+1}=r/|q|^{n+2}\). Then

$$\begin{aligned} \left| \frac{f(qx)-f(x)}{x}\right| \le \frac{|q|^{n+2}}{r}\Vert f\Vert _n\left( \frac{1}{d_n(|qx|)^n}+\frac{1}{d_n(|x|)^n}\right) \le \frac{2 |q|^{n+2} \Vert f\Vert _n}{r d_{n+1}(|x|)^n}, \end{aligned}$$
(17)

thanks to Lemma 4(2) and (3). Thus, we find that

$$\begin{aligned} \left| \frac{f(qx)-f(x)}{x}\right| d_{n+1}(|x|)^{n+1}&\le 2\frac{|q|^{n+2}}{r}\Vert f\Vert _n d_{n+1}(|x|)\\&\le 2|q|^{n+2}\Vert f\Vert _n \frac{(r-\rho )}{r}=2|q|^{n+1}\Vert f\Vert _n(|q|-1). \end{aligned}$$

If \(|x|\le \rho /|q|^{n+1}\), using the Maximum Modulus Principle and (17) we see that

$$\begin{aligned} \left| \frac{f(qx)-f(x)}{x}\right|&\le \sup _{|x|=\rho /|q|^{n+1}} \left| \frac{f(qx)-f(x)}{x}\right| \le \frac{2 |q|^{n+2} \Vert f\Vert _n}{r (r-\rho )^n}. \end{aligned}$$

Therefore,

$$\begin{aligned} \left| \frac{f(qx)-f(x)}{x}\right| d_{n+1}(|x|)^{n+1}&\le 2|q|^{n+1}\Vert f\Vert _n(|q|-1). \end{aligned}$$

Since \(|q|-1\le |q-1|\), these inequalities establish (2).

Finally, to prove (3) simply note that by definition of \(\Vert f\Vert _n\) we have

$$\begin{aligned} |f(qx)|\le \frac{\Vert f\Vert _n}{d_n(|qx|)^{n}}=\frac{\Vert f\Vert _n}{d_{n+1}(|x|)^{n}},\qquad \text { for } |x|<r/|q|^{n+1}. \end{aligned}$$

In this way, \(|f(qx)| d_{n+1}(|x|)^{n+1}\le \Vert f\Vert _n d_{n+1}(|x|)<\Vert f\Vert _n(r-\rho )\) as required. \(\square \)

Remark

Fix \(f\in {\mathcal {O}}_b(D_r)\), \(z=(z_1,\dots ,z_N)\in {\mathcal {O}}_b(D_r)^N\), \(A=(a_{ij})\in {\mathcal {O}}_b(D_r)^{N\times N}\), and set

$$\begin{aligned} \Vert z\Vert _n=\max _{1\le j\le N} \Vert z_j\Vert _n,\qquad \Vert A\Vert _n=\max _{1\le i\le N} \sum _{j=1}^N \Vert a_{i,j}\Vert _n. \end{aligned}$$

The previous lemma shows that

$$\begin{aligned} \Vert f\cdot z\Vert _{n+m}&\le \Vert f\Vert _n \Vert z\Vert _m, \quad \Vert A\cdot z\Vert _{n+m}\le \Vert A\Vert _n \Vert z\Vert _m,\\ \Vert d_qz\Vert _{n+1}&\le 2|q|^{n+1} \Vert z\Vert _n,\qquad \Vert \sigma _q z\Vert _{n+1}\le r\left( 1-\frac{1}{|q|}\right) \Vert z\Vert _n, \end{aligned}$$

for all n, m.

5 Proof of the main results

This section is devoted to prove Theorems 1 and 2 on the divergence rate of the formal solutions of (4), (5), and (6), respectively, using majorant series. For expansions in \(\epsilon \), we use the q-Nagumo norms to control \(d_{q,x}\) and \(\sigma _{q,x}\).

The proofs are the same in all cases. The structure has been used successfully in the differential setting, see e.g., [10, 11, 20]. We will apply the following steps:

  1. 1.

    Apply rank reduction in \(\epsilon \) to assume \( \alpha =1\).

  2. 2.

    Find the formal solution \({\hat{y}}\) such that \({\hat{y}}(0,0)=0\), starting by the first coefficient (as a function of x or \(\epsilon \)) using the Implicit Function Theorem. Then, set the recurrences to determine the other terms.

  3. 3.

    Using the supremum norm in \(D_r\) or \(D_{r/|q|^n}\) (\({\hat{y}}\) as power series in x) or the q-Nagumo norms (\({\hat{y}}\) as power series in \(\epsilon \)), find a system of inequalities for the norms \(z_n\ge 0\) of the coefficients. Then, divide them by an adequate log-convex sequence \(\{M_n\}_{n\ge 1}\) to find a sequence of inequalities for \(z_n/M_n\).

  4. 4.

    Associate an analytic problem having a unique convergent series solution \({\hat{w}}=\sum w_n \tau ^n\) that majorises \(\sum \frac{z_n}{M_n}\tau ^n\), i.e., \(z_n/M_n\le w_n\), for all n.

Except for Step 3 where \(M_n\) is chosen, the arguments are similar in all cases. Therefore, we will only write one in detail, namely, analyzing the coefficients of \({\hat{y}}\) as a power series in x. For the others, we will only indicate the necessary modifications.

Proof

Solution as a power series in  x.  Case \(p>0\).

Step 1: Rank reduction. We can perform rank reduction on the systems (4) and (5), in exactly the same way as for ODEs, to reduce this to the case \( \alpha =p=1\). However, to preserve the nature of the problem we only reduce rank in \(\epsilon \), see Remark at page 17. Thus, we write

$$\begin{aligned} y(x,\epsilon )=\sum _{j=0}^{ \alpha -1} y_j(x,\epsilon ^ \alpha ) \epsilon ^j,\qquad w(x,\eta )=(y_0(x,\eta ),y_1(x,\eta ),\dots ,y_{ \alpha -1}(x,\eta )).\end{aligned}$$
(18)

Notice that \(\epsilon ^ \alpha x^{p+1}d_{q,x}y(x,\epsilon )=\sum _{j=0}^{ \alpha -1} \eta x^{p+1}d_{q,x}y_j(x,\eta ) \epsilon ^j\) and \(\epsilon ^ \alpha x^{p}\sigma _{q,x}y(x,\epsilon )=\sum _{j=0}^{ \alpha -1} \eta x^{p}\sigma _{q,x}y_j(x,\eta ) \epsilon ^j\). Also, expanding

$$\begin{aligned} F(x,\epsilon ,y)=\sum _{l=0}^{ \alpha -1} F_l\left( x,\epsilon ^ \alpha ,\sum _{k=0}^{ \alpha -1} y_k(x,\epsilon ^ \alpha ) \epsilon ^k\right) \epsilon ^l = \sum _{j=0}^{ \alpha -1} {\widetilde{F}}_j(x,\eta ,w) \epsilon ^j, \end{aligned}$$

and setting \(G(x,\eta ,w)=({\widetilde{F}}_0,,\dots ,{\widetilde{F}}_{ \alpha -1})\), it follows that \(DG_w(0,0,0)\) is a block lower-triangular matrix of size \(N \alpha \) having all diagonal blocks equal to \(DF_y(0,0,0)\), see [7, Section 4] for details in the differential case. Thus, this matrix is invertible. The new systems, of dimension \(N \alpha \), take the form

$$\begin{aligned} \eta x^{p+1} d_qw=G(x,\eta ,w),\qquad \eta x^{p} \sigma _{q}w=G(x,\eta ,w), \end{aligned}$$

and have the same structure as (4) and (5), but now \( \alpha =1\).

Step 2: Existence. We can assume now that (4) and (5) have the form

$$\begin{aligned} \epsilon x^{p+1}d_{q}(y)(x,\epsilon )=F(x,\epsilon ,y),\qquad \epsilon x^{p}\sigma _{q}(y)(x,\epsilon )=F(x,\epsilon ,y),\end{aligned}$$
(19)

where \(y\in {\mathbb {C}}^N\), F is holomorphic near \((0,0,0)\in {\mathbb {C}}\times {\mathbb {C}}\times {\mathbb {C}}^N\), \(F(0,0,0)=0\), and \(DF_y(0,0,0)\) is invertible.

The existence and uniqueness of a solution \({\hat{y}}\in {\mathbb {C}}[[x,\epsilon ]]^N\) such that \({\hat{y}}(0,0)=0\) follows by replacing \({\hat{y}}\) into (19) and solving the coefficients recursively. This can be done thanks to the hypothesis of \(DF_y(0,0,0)\) being invertible. Here we are interested in the coefficients \(y_n(\epsilon )\) of \({\hat{y}}\) when written in the first form of (3). To establish the recurrences that define them, let us write

$$\begin{aligned} {F}(x,\epsilon ,{y})=b(x,\epsilon )+A(x,\epsilon ){y}+H(x,\epsilon ,y),\qquad H(x,\epsilon ,y)=\sum _{I\in {\mathbb {N}}^N, |I|\ge 2} A_I(x,\epsilon ) {y}^I, \end{aligned}$$

as a convergent power series in \({ y}\) with coefficients \(A(x,\epsilon )=\sum _{n=0}^\infty A_{n*}(\epsilon )x^n\), and analogously for \(b(x,\epsilon )\) and each \(A_I(x,\epsilon )\). Thus, there is \(r>0\) such that \(A_{n*}\in {\mathcal {O}}_b(D_r)^{N\times N}\) and \(b_{n*}, A_{I,n*}\in {\mathcal {O}}_b(D_r)^N\), for all \(n\in {\mathbb {N}}\). Note that we are using the notation \(y^{I}=y_1^{i_1}\cdots y_N^{i_N}\), where \(I=(i_1,\dots ,i_N)\in {\mathbb {N}}^N\).

The first coefficient \(y_0(\epsilon )\), which must satisfy \(y_0(0)=0\), is determined by solving the implicit equation

$$\begin{aligned} F(0,\epsilon ,y_0(\epsilon ))=0. \end{aligned}$$

This has a unique analytic solution \(y_0(\epsilon )\in {\mathcal {O}}_b(D_{r'})^N\), for some \(r'>0\), via the Implicit Function Theorem since \(F(0,0,0)=0\) and \(DF_y(0,0,0)=A(0,0)\) is invertible. Reducing r if necessary, assume \(r'=r\) and \(A(0,\epsilon )=A_{0*}(\epsilon )\) invertible and bounded for all \(|\epsilon |<r\).

After the change of variables \(y\mapsto y-y_0(\epsilon )\) in the initial equation (19), we can assume now that \(y_0(\epsilon )=0\). Thus we obtain a similar q-difference equation such that \(F(0,\epsilon ,0)=b_{0*}(\epsilon )=0\). Now, after replacing \({\hat{y}}=({\hat{y}}_1,\dots ,{\hat{y}}_N)=\sum _{n=1}^\infty y_n(\epsilon ) x^n\), where \(y_n(\epsilon )=( y_{1,n}(\epsilon ),\dots ,y_{N,n}(\epsilon ))\), into (19), we obtain the recurrence

$$\begin{aligned} \epsilon [n-p]_q y_{n-p}(\epsilon )&=b_{n*}(\epsilon )+\sum _{j=1}^n A_{n-j*}(\epsilon ) y_{j}(\epsilon )\nonumber \\&\quad +\sum _{k=2}^n \sum _{*_k} \prod _{\begin{array}{c} {1\le l\le N}\\ {1\le j\le i_l} \end{array}} y_{l,n_{l,j}}(\epsilon )\,\cdot A_{I,m*}(\epsilon ),\quad n\ge 1, \end{aligned}$$
(20)

for the first system. The same recurrence holds for the second system with \([n-p]_{q}\) replaced by \(q^{n-p}\). The inner sum indicated with \((*_k)\) is taken over all \(I=(i_1,\dots ,i_N)\in {\mathbb {N}}^N\) such that \(|I|=k\), m satisfying \(0\le m\le n-k\), and \(n_{l,j}\ge 1\) such that \(n_{1,1}+\cdots +n_{1,i_1}+\cdots +n_{N,1}+\cdots +n_{N,i_N}+m=n\). Note in particular that \(n_{l,j} < n-m\le n\), thus no component of \({\hat{y}}_n\) appears in the former sum. Finally, the left-side of the equation is understood as zero for \(n<p\).

Since \(A_{0*}(\epsilon )\) is invertible for \(|\epsilon |<r\), \(y_{n}(\epsilon )\) is uniquely determined by (20) and is analytic and bounded on \(D_r\). The same holds for the system involving \(\sigma _{q}\).

Step 3: Estimates for \(y_n(\epsilon )\). We use the supremum norm \(\Vert \cdot \Vert \), i.e., the q-Nagumo norms of order 0 as remarked at page 10. Let \(c=\Vert A_{0*}^{-1}\Vert >0\), \( \alpha _n=\Vert A_{n*}\Vert \), \(\beta _n=\Vert b_{n*}\Vert \), and \(\gamma _{I,m}=\Vert A_{I,m*}\Vert \). For the first system in (19) it follows from (20) that \(z_n:=\Vert y_n\Vert \) satisfies the inequalities

$$\begin{aligned} \frac{z_n}{c}\le \beta _n+ r[n-p]_{|q|} z_{n-p}+\sum _{j=1}^{n-1} \alpha _{n-j} z_j+\sum _{k=2}^n \sum _{*_k} \prod _{\begin{array}{c} {1\le l\le N}\\ {1\le j\le i_l} \end{array}} z_{n_{l,j}}\,\cdot \gamma _{I,m}. \end{aligned}$$
(21)

Note that we used that \(\Vert y_{l,n_{l,j}}\Vert \le \Vert y_{n_{l,j}}\Vert =z_{n_{l,j}}\). Now, we choose the sequence

$$\begin{aligned} M_n=([n]^{!}_{|q|})^{1/p}, \end{aligned}$$

which is log-convex, see Remark at page 6 and (14). Recalling (12) we find that

$$\begin{aligned} \frac{z_n}{c\cdot M_n}\le \frac{\beta _n}{M_n}+r \frac{z_{n-p}}{M_{n-p}}+\sum _{j=1}^{n-1} \frac{ \alpha _{n-j}}{M_{n-j}} \frac{z_j}{M_j}+\sum _{k=2}^n \sum _{*_k} \prod _{\begin{array}{c} {1\le l\le N}\\ {1\le j\le i_l} \end{array}} \frac{z_{n_{l,j}}}{M_{n_{l,j}}}\,\cdot \frac{\gamma _{I,m}}{M_m}. \end{aligned}$$
(22)

For the second system we use the corresponding inequality to (20), to obtain that \(z_n=\Vert y_n\Vert \) satisfies (21) with \([n-p]_{|q|}\) replaced by \(|q|^{n-p}\). Then we divide by

$$\begin{aligned} M_n=|q|^{n^2/2p}, \end{aligned}$$

and noticing that \(|q|^{n-p}/M_n\le 1/M_{n-p}\), we arrive again to (22).

Step 4: The majorized problem. Define recursively \(w_n\) by \(w_1=\Vert y_1\Vert \) and

$$\begin{aligned} \frac{w_n}{c}= \frac{\beta _n}{M_n} + r w_{n-p}+\sum _{j=1}^{n-1} \frac{ \alpha _{n-j}}{M_{n-j}} w_j+\sum _{k=2}^n \sum _{*_k} \prod _{\begin{array}{c} {1\le l\le N}\\ {1\le j\le i_l} \end{array}} w_{n_{l,j}} \frac{\gamma _{I,m}}{M_m}. \end{aligned}$$
(23)

It follows by induction that \(z_n/M_n\le w_n\), for all \(n\ge 1\). Recursion (23) is equivalent to assert that \({\hat{w}}(\tau ):=\sum _{n\ge 1}w_n \tau ^n\) satisfies

$$\begin{aligned} \frac{1}{c}{\hat{w}}(\tau ) = {\widetilde{b}}(\tau )+r\tau ^p {\hat{w}}(\tau )+{\widetilde{A}}(\tau ){\hat{w}}(\tau )+\sum _{|I|\ge 2}{\widetilde{A}}_I(\tau ){\hat{w}}^{|I|}, \end{aligned}$$
(24)

where

$$\begin{aligned} {\widetilde{b}}(\tau )=\sum _{n=1}^\infty \frac{\beta _n}{M_n} \tau ^n,\quad {\widetilde{A}}(\tau )=\sum _{n=1}^\infty \frac{ \alpha _{n}}{M_n} \tau ^n,\qquad {\widetilde{A}}_I(\tau )=\sum _{m=0}^\infty \frac{\gamma _{I,m}}{M_m} \tau ^m, \end{aligned}$$

define analytic (in this case, actually entire) functions of \(\tau \). Consider the map

$$\begin{aligned} H(\tau ,w)={\widetilde{b}}(\tau )+\left( r\tau ^p+{\widetilde{A}}(\tau )-\frac{1}{c}\right) w+\sum _{k=2}^\infty \left( \sum _{|I|=k} {\widetilde{A}}_{I}(\tau )\right) w^k, \end{aligned}$$

which is analytic at \((\tau ,w)=(0,0)\), due to estimates of the form

$$\begin{aligned} \Vert A_I\Vert \le K\delta ^{|I|},\qquad |I|\ge 2,\end{aligned}$$
(25)

for some constants \(K,\delta >0\). They hold thanks to the analyticity of F at the origin. Therefore, we can apply the Implicit Function Theorem to H since \(\frac{\partial H}{\partial w}(0,0)=-\frac{1}{c}\ne 0\). Thus we find a unique analytic solution \({\widetilde{w}}(\tau )\) of \(H(\tau ,{\widetilde{w}}(\tau ))=0\) and \({\widetilde{w}}(0)=0\). But \({\hat{w}}={\widetilde{w}}\), since both are formal solutions, and those are unique. In conclusion, \({\hat{w}}\in {\mathbb {C}}\{\tau \}\). Thus \(\hat{{y}}\) satisfies

$$\begin{aligned} \Vert y_n\Vert \le C L^n ([n]^{!}_{|q|})^{1/p},\qquad \Vert y_n\Vert \le CL^n |q|^{n^2/2p}, \end{aligned}$$

respectively for each system, and for some constants \(C,L>0\). Returning to the variable \(\epsilon \) before rank reduction, the only effect in the previous bounds is to change the radius \(|\eta |<r\) by \(|\epsilon |<r^{1/ \alpha }\). This concludes the case for x and \(p>0\).

Solution as a power series in  x. Case  \(p=0\). Here the first equation in (19) has the form \(\epsilon x d_q(y)(x,\epsilon )=F(x,\epsilon ,y)\). Solving for y as a power series in x, and assuming already that \(y_0(\epsilon )=0\), the coefficient \(y_n(\epsilon )\) is now determined by

$$\begin{aligned} y_{n}(\epsilon )=\left( \epsilon [n]_qI_N-A_{0*}(\epsilon )\right) ^{-1}\left[ b_{n*}(\epsilon )+\sum _{j=1}^{n-1} A_{n-j*}(\epsilon ) y_{j}(\epsilon )+\cdots \right] ,\quad n\ge 1. \end{aligned}$$
(26)

In this case, the reduction on the radius on \(\epsilon \) comes from \(\left( \epsilon [n]_q I_N-A_{0*}(\epsilon )\right) ^{-1}\). Indeed, write \(A_0=A_{0*}(0)\) and choose \(0<r<1/4\Vert A_{0}^{-1}\Vert \) with \(\Vert A_{0}-A_{0*}(\epsilon )\Vert <1/4\Vert A_{0}^{-1}\Vert \), for all \(|\epsilon |\le r\). Then, if \(|\epsilon |\le r/|[n]_q|\), it follows that

$$\begin{aligned} \Vert A_{0}^{-1}\left( \epsilon [n]_q I_N+A_{0}-A_{0*}(\epsilon )\right) \Vert \le \frac{1}{2}<1, \end{aligned}$$

and using the inequality \(\Vert (I-B)^{-1}\Vert \le \frac{1}{1-\Vert B\Vert }\), valid for \(\Vert B\Vert <1\) (Neumann series), for any matrix norm, we find that

$$\begin{aligned} \Vert \left( \epsilon [n]_q I_N{-}A_{0*}(\epsilon )\right) ^{-1}\Vert&= \Vert \left( I_N-A_0^{-1}\left( \epsilon [n]_q I_N+A_{0}-A_{0*}(\epsilon )\right) \right) ^{-1}\cdot (-A_0)^{-1}\Vert \\&\le 2\Vert A_0^{-1}\Vert =c. \end{aligned}$$

If \(z_n=\sup _{|\epsilon |\le r/|[n]_q|} |y_n(\epsilon )|\), \( \alpha _n=\sup _{|\epsilon |\le r/|[n]_q|} \Vert A_{*n}(\epsilon )\Vert \), \(\beta _n=\cdots \), we find

$$\begin{aligned} \frac{z_n}{c}\le \beta _n+\sum _{j=1}^{n-1} \alpha _{n-j} z_j+\cdots . \end{aligned}$$
(27)

It follows that

$$\begin{aligned} \sup _{|\epsilon |\le r/|[n]_q|} |y_n(\epsilon )|\le CA^n, \end{aligned}$$

for some constants \(C,A>0\), and all n, as needed. Note we can interchange the terms \([n]_q\) and \(q^n\) in this supremum by recalling (8) and reducing r if necessary.

For the second equation in (19), \(y_n(\epsilon )\) is determined by (26) with \(q^n\) instead of \([n]_q\). As before, the matrix \(\left( \epsilon q^n I_N-A_{0*}(\epsilon )\right) ^{-1}\) can be uniformly bounded for \(|\epsilon |\le r/|q|^n\), for an adequate \(r>0\). Letting \(z_n=\sup _{|\epsilon |\le r/|q|^n} |y_n(\epsilon )|\), \( \alpha _n=\sup _{|\epsilon |\le r/|q|^n} \Vert A_{*n}(\epsilon )\Vert \), \(\beta _n=\cdots \), we arrive again at (27) and to the desired bounds.

To conclude, note that for general \( \alpha >1\), when we return to the variable \(\epsilon \), the condition \(|\eta |\le r/|q|^n\) means that \(|\epsilon |\le r'/|q|^{\frac{n}{ \alpha }}\) where \(r'=r^{1/ \alpha }\), since \(\eta =\epsilon ^ \alpha \).

Solution as a power series in \(\epsilon \). Case \(p\ge 0\). Consider the system in (19) with \( \alpha =1\) and search for the coefficients \(u_n(x)\) of \({\hat{y}}\), as power series in \(\epsilon \). The coefficient \(u_0(x)\) with \(u_0(0)=0\) is determined by solving \(F(x,0,u_0(x))=0\). After the change of variables \(y\mapsto y-u_0(x)\) in equation (19), we obtain a similar one with \(F(x,0,0)=b_{*0}(x)=0\). As before, replacing \({\hat{y}}=({\hat{y}}_1,\dots ,{\hat{y}}_N)\), \({\hat{y}}_j=\sum _{n=1}^\infty u_{j,n}(x) \epsilon ^n\), into the first system in (19), we obtain the family of q-difference equations

$$\begin{aligned} x^{p+1}d_q(u_{n-1})=&b_{*n}+\sum _{j=1}^n A_{*n-j}u_{j} +\sum _{k=2}^n \sum _{*_k} \prod _{\begin{array}{c} {1\le l\le N}\\ {1\le j\le i_l} \end{array}} u_{l,n_{l,j}}\,\cdot A_{*I,m}, \end{aligned}$$
(28)

where the sum \((*_k)\) has the same structure as before. For the second system we obtain (28) with \(x^{p}\sigma _q(u_{n-1})\) instead of \(x^{p+1}d_q(u_{n-1})\).

Here \(A(x,\epsilon )=\sum _{n=0}^\infty A_{*n}(x)\epsilon ^n\) and similarly for the other terms. The radius \(r>0\) is such that \(A_{*n}\in {\mathcal {O}}_b(D_r)^{N\times N}\), \(b_{*n}, A_{*I,n}\in {\mathcal {O}}_b(D_r)^N\), and \(A_{*0}(x)\) is invertible and bounded for \(|x|<r\). Thus, in both cases the coefficient \(u_n\) are uniquely determined by the previous terms. Moreover, since the operators \(d_q\) and \(\sigma _q\) reduce the radius of convergence by a factor of q, \(u_n\in {\mathcal {O}}_b(D_{r/|q|^n})\).

For Step 3, let \(c=\Vert A_{*0}^{-1}\Vert _0\), \(z_n=\Vert u_{n}\Vert _n\), \( \alpha _n=\Vert A_{*n}\Vert _n\), \(\beta _n=\Vert b_{*n}\Vert _n\), and \(\gamma _{I,m}=\Vert A_{*I,m}\Vert _m\). Equation (28) and the properties of the q-Nagumo norms developed in Sect. 4 establish that

$$\begin{aligned} \frac{z_n}{c}\le \beta _n+2 r^{p+1} z_{n-1}+\sum _{j=1}^{n-1} \alpha _{n-j}z_j+\sum _{k=2}^n \sum _{*_k} \prod _{\begin{array}{c} {1\le l\le N}\\ {1\le j\le i_l} \end{array}} z_{n_{l,j}}\,\cdot \gamma _{I,m}. \end{aligned}$$
(29)

In fact, the q-derivative \(d_q\) is controlled by the inequality

$$\begin{aligned} \Vert x^{p+1}d_q(u_{n-1})\Vert _n\le \sup _{|x|\le \frac{r}{|q|^n}} |x^{p+1}|\cdot \Vert d_q (u_{n-1})\Vert _n\le 2r^{p+1} \Vert u_{n-1}\Vert _{n-1}, \end{aligned}$$

valid for \(p\ge 0\), as follows from the second statement in Lemma 5. In the case of the operator \(\sigma _q\), a direct application of the third statement in Lemma 5 yields

$$\begin{aligned} \Vert x^{p}\sigma _q(u_{n-1})\Vert _n\le \sup _{|x|\le \frac{r}{|q|^n}} |x^{p}|\cdot \Vert \sigma _q (u_{n-1})\Vert _n\le r^{p}\cdot r \Vert u_{n-1}\Vert _{n-1}, \end{aligned}$$

thus arriving to the same inequalities for the \(z_n\).

Now we choose \(M_n=1\) to conclude in the same way as in previous cases, that \(\Vert u_n\Vert _n\le DB^n,\) for some constants \(B,D>0\). The definition of the norms leads to

$$\begin{aligned} |u_{n}(x)|\le \frac{\Vert u_n\Vert _n}{d_n(|x|)^n}\le \frac{DB^n }{(r-\rho )^n} = D L^n, \end{aligned}$$

where \(L=B|q|/r(|q|-1)\) and \(|x|\le {\rho }/{|q|^n}={r}/{|q|^{n+1}}\), as required.

Finally, we establish the case \( \alpha >1\). If we return to the original variable \(\epsilon \) before rank reduction, \(y(x,\epsilon )=\sum _{j=0}^{ \alpha -1} y_j(x,\epsilon ^ \alpha ) \epsilon ^j=\sum _{n=0}^\infty u_n(x) \epsilon ^n\), and \(y_j(x,\eta )=\sum _{k=0}^\infty u_{j,k}(x) \eta ^k\), we have that

$$\begin{aligned} u_{ \alpha k+j}(x)=u_{j,k}(x),\qquad j=0,1,\dots , \alpha -1, k\ge 0. \end{aligned}$$

The proof has shown that \(|u_{j,k}(x)|\le D L^k\), for all \(|x|\le \rho /|q|^{k}\), and k. If \(n\ge 0\) is divided by \( \alpha \) and \(n= \alpha k+j\), then \(k=\lfloor n/ \alpha \rfloor \). From here we conclude that

$$\begin{aligned} \sup _{ |x|\le {\rho }/{|q|^{\lfloor \frac{n}{ \alpha }\rfloor }} } |u_n(x)|\le C'L'^n, \end{aligned}$$

for some \(C'=C'(q)\), \(L'=L'(q)>0\). In general, the value \(r/|q|^{\lfloor n/ \alpha \rfloor }\) means that the radius of the domain of the coefficients \(u_n\) is reduced by a factor of |q| every \( \alpha \) steps, i.e., \(u_{k \alpha }, u_{k \alpha +1},\dots , u_{k \alpha + \alpha -1}\in {\mathcal {O}}(D_{r/|q|^k})\), for all k. \(\square \)

Finally, we conclude by proving Theorem 2.

Proof of Theorem 2

We proceed as in Theorem 1 searching for a solution as a power series in \(\epsilon \). In this case, after rank reduction, i.e., assuming \( \alpha =1\), if we search for a solution of (6) of the form \({\hat{y}}=\sum _{n=0}^\infty u_n(x) \epsilon ^n\), we will arrive at recurrence (28) with \(p=-1\). The induced inequalities for \(z_n=\Vert u_n\Vert _n\) are

$$\begin{aligned} \frac{z_n}{c}\le \beta _n+2 |q|^n z_{n-1}+\sum _{j=1}^{n-1} \alpha _{n-j}z_j+\cdots . \end{aligned}$$

In this case we choose

$$\begin{aligned} M_n=|q|^n\cdot |q|^{n^2/2}, \end{aligned}$$

which is log-convex since \(M_n/M_{n-1}=|q|^{n+\frac{1}{2}}\) is increasing in n, and satisfies \(|q|^n/M_n\le 1/M_{n-1}\). Therefore, dividing the previous inequality by \(M_n\) and proceeding as before, we find that

$$\begin{aligned} \Vert u_n\Vert _n\le DB^n |q|^n |q|^{n^2/2}, \end{aligned}$$

for some \(B,D>0\). Recalling the definition of q-Nagumo norms, this means that

$$\begin{aligned} |u_n(x)|\le DL^n |q|^{n^2/2}, \end{aligned}$$

for \(|x|\le \rho /|q|^{n}=r/|q|^{n+1}\), where \(L=B|q|^2/r(|q|-1)>0\).

Finally, for general \( \alpha >1\), using the same notation that in the last paragraph of the previous proof, in this case \(\sup _{|x|\le \rho /|q|^k} |u_{j,k}(x)|\le DL^k |q|^{k^2/2}\). Therefore, if \(n= \alpha k+j\), and \(k=\lfloor n/ \alpha \rfloor \), we find that

$$\begin{aligned} \sup _{|x|\le \rho /|q|^{\lfloor n/ \alpha \rfloor }} |u_n(x)|\le DL^{n/ \alpha } |q|^{(n/ \alpha )^2/2}, \end{aligned}$$

as required. Here we have assumed that \(L>1\). This concludes the proof. \(\square \)

Remark

If we reduce rank in x and \(\epsilon \), we should consider the decomposition

$$\begin{aligned} y(x,\epsilon )=\sum _{\begin{array}{c} {0\le l<p}\\ {0\le j< \alpha } \end{array}} y_{l,j}(z,\eta ) x^l\epsilon ^j,\qquad \text { where } z=x^p, \eta =\epsilon ^ \alpha . \end{aligned}$$

Computing \(d_{q,x}(y)\), it follows using (10) and (7) that

$$\begin{aligned} \epsilon ^{ \alpha } x^{p+1}d_{q,x}(y)=\sum _{\begin{array}{c} {0\le l<p}\\ {0\le j< \alpha } \end{array}} \left( [p]_q q^l \eta z^2 d_{q^p,z}(y_{l,j})(z,\eta ) +[l]_q \eta z y_{l,j}(z,\eta )\right) x^l\epsilon ^j. \end{aligned}$$

Then \(w(z,\eta )=(y_{l,j})_{0\le l<p, 0\le j< \alpha }\), written in lexicographical order, satisfies

$$\begin{aligned} \eta z^2 d_{q^p,z}w=G(z,\eta ,w), \end{aligned}$$

where \(DG_w(0,0,0)=[p]_q^{-1}D_q{\widetilde{A}}\), \(D_q=\text {diag}(I_{ \alpha N}, q^{-1} I_{ \alpha N},\dots ,q^{-(p-1)} I_{ \alpha N})\), and \({\widetilde{A}}\) is a block lower-triangular matrix having A(0, 0) as diagonal blocks. Apart from the linear part, G also contains a term depending on q, namely, \(\eta z[p]_q^{-1}D_qM_q\), where

$$\begin{aligned} M_q=\text {diag}(0, [1]_q I_{ \alpha N},\dots ,[p-1]_q I_{ \alpha N}). \end{aligned}$$

Therefore, this reduction would force to change the nature of the problem by introducing q on the non-q-difference part of the equation.

6 Examples

This section is devoted to give examples of our results. We remark that in only few cases the formal solution can be easily computed.

Remark

In the following examples we impose the condition \(F(0,0,0)=0\) (achieved by a change of variables \(w=y-a\), a constant), as it is required in Theorems 1 and 2. We note that in our results we request that \({\hat{y}}(0,0)=0\). Otherwise the uniqueness of a power series solution might not be achieved. For instance, for the scalar equation

$$\begin{aligned} \epsilon ^ \alpha x^{p+1}d_{q,x}(y)(x,\epsilon )=F(x,\epsilon ,y)=c(\epsilon )(y-\lambda _1(\epsilon ))\cdots (y-\lambda _k(\epsilon )) y, \end{aligned}$$

we have the solutions \(y=0\) and \(y=\lambda _j(\epsilon )\), \(j=1,\dots ,k\), which in general are different. Since \(DF_y(0,0,0)=(-1)^k c(0)\lambda _1(0)\cdots \lambda _k(0)\) is required to be invertible, the unique formal power series solution \({\hat{y}}(x,\epsilon )\) of this equation satisfying \({\hat{y}}(0,0)=0\) is \({\hat{y}}=0\).

An analogous situation holds for the equation

$$\begin{aligned} \epsilon ^ \alpha x^p \sigma _q y=F(x,\epsilon ,y)=y(y-\lambda (\epsilon )+\epsilon ^ \alpha x^p ), \end{aligned}$$

having the two solutions \(y=0\) and \(y=\lambda (\epsilon )\). Since \(DF_y(0,0,0)=-\lambda (0)\), the unique formal solution provided by our results is \({\hat{y}}=0\).

Example 3

(\(\sigma _q\), case \(p=0\)) Consider the problem

$$\begin{aligned} \epsilon ^ \alpha \sigma _{q,x}y=A(x)y-b(x), \end{aligned}$$

where \( \alpha \in {\mathbb {N}}^+\), \(y\in {\mathbb {C}}^N\) and A(0) is an invertible matrix. It follows that

$$\begin{aligned} {\hat{y}}_q(x,\epsilon )=\sum _{m=0}^\infty \left[ A(q^m x)\cdots A(qx)A(x)\right] ^{-1} b(q^m x) \epsilon ^{ \alpha m}, \end{aligned}$$

is the unique formal power series solution of the problem. Although \(F(x,\epsilon ,y)=A(x)y-b(x)\) only satisfies that \(F(0,0,0)=0\) when \(b(0)=0\), we can make the change of variables \(w=y-{\hat{y}}_q(0,0)=y-A(0)^{-1}b(0)\) to obtain the equivalent system

$$\begin{aligned} \epsilon ^ \alpha \sigma _{q,x}w=G(x,\epsilon ,w)=A(x)w-{\widetilde{b}}(x,\epsilon ),\quad {\widetilde{b}}(x,\epsilon )=b(x)+(\epsilon ^ \alpha I_N-A(x)) A(0)^{-1}b(0), \end{aligned}$$

for which \(G(0,0,0)=0\). The unique formal power series solution of this system with \({\hat{w}}_q(0,0)=0\) is \({\hat{w}}_q(x,\epsilon )={\hat{y}}_q(x,\epsilon )-{\hat{y}}_q(0,0)\). We see that \({\hat{y}}_q, {\hat{w}}_q\in {\mathcal {O}}_{0}^{q^{1/ \alpha }}\), confirming Theorem 1(2) for \(\sigma _q\). In the limit \(q\rightarrow 1\), \({\hat{y}}_q\) reduces to \({\hat{y}}_1=\left[ A(x)-\epsilon ^ \alpha I_N\right] ^{-1} \cdot b(x)\), which is of course the unique analytic solution of the limit problem \(\epsilon ^ \alpha y(x,\epsilon )=A(x)y(x,\epsilon )-b(x)\). A particular interesting case is the equation

$$\begin{aligned} \epsilon ^ \alpha \sigma _{q,x}y=(1-x)y-1, \end{aligned}$$

generating the solution

$$\begin{aligned} {\hat{y}}_q(x,\epsilon )=\sum _{m=0}^\infty \frac{\epsilon ^{ \alpha m}}{(x;q)_{m+1}}=\sum _{n,m=0}^\infty \left[ \begin{array}{l}{n+m}\\ {m}\end{array}\right] _{q} x^n \epsilon ^{ \alpha m}=\sum _{n=0}^\infty \frac{x^n}{(\epsilon ^ \alpha ;q)_{n+1}}. \end{aligned}$$

Note we used Heine’s binomial formula, see [18, p. 28]. Since \(\left[ \begin{array}{l}{n+m}\\ {m}\end{array}\right] _{q}\) is a monic polynomial in q of degree mn, we have the precise bounds given in Theorem 1(2) for \(\sigma _q\). Additionally, the series reduces to \({\hat{y}}_1=(1-x-\epsilon ^ \alpha )^{-1}\) for \(q=1\).

We also highlight the case \(A(x)=I_N\) and \(b(x)=\sum _{n=0}^\infty a_n x^n\in {\mathbb {C}}\{x\}\) for which

$$\begin{aligned} {\hat{y}}_q(x,\epsilon )=\sum _{m=0}^\infty b(q^m x) \epsilon ^{ \alpha m}=\sum _{n,m=0}^\infty a_n q^{nm} x^n\epsilon ^{ \alpha m}=\sum _{n=0}^\infty \frac{a_n x^n}{1-q^n \epsilon ^ \alpha }. \end{aligned}$$

If \( \alpha =1\) and \(b(x)=(1-x)^{-1}\), we recover the series \(M(x,\epsilon )\) of Example 2.

Example 4

(\(d_q\), case \(p=0\)) Consider the problem

$$\begin{aligned} \epsilon ^ \alpha x d_{q,x}y=y-f_0(x), \end{aligned}$$

with \(f_0(x)=\sum _{n=0}^\infty a_n x^n\in {\mathbb {C}}\{x\}\). Note we achieve the hypotheses of Theorem 1 after the translation \(w=y-a_0\). Solving directly for \({\hat{y}}=\sum _{m=0}^{\infty } u_m(x)\epsilon ^{ \alpha m}\) we see that \(u_m(x)=xd_q(u_{m-1})=\cdots =(xd_q)^m(f_0)\). Therefore,

$$\begin{aligned} {\hat{y}}=a_0+\sum _{n\ge 1,m\ge 0} a_n [n]_q^m x^n \epsilon ^{ \alpha m}=a_0+\sum _{n=1}^\infty \frac{a_n x^n}{1-[n]_q \epsilon ^ \alpha } \in {\mathcal {O}}_0^{q^{1/ \alpha }}, \end{aligned}$$

as claimed by Theorem 1(2). The same conclusion can be achieved by writing \({\hat{y}}=\sum _{n=0}^\infty y_n(\epsilon ) x^n\), and solving the recurrence \([n]_q \epsilon ^ \alpha y_n(\epsilon )=y_n(\epsilon )-a_n\).

Example 5

(\(\sigma _q\), \(p>0\), \( \alpha =1\)) Consider the equation

$$\begin{aligned} \epsilon x^p\sigma _{q,x}y=y-f(x,\epsilon ), \end{aligned}$$

where \(f(x,\epsilon )=\sum _{m=0}^\infty f_m(x) \epsilon ^m\in {\mathbb {C}}\{x,\epsilon \}\) and \(f(0,0)=0\). It follows that \({\hat{y}}=\sum _{m=0}^\infty u_m(x) \epsilon ^m\) satisfies the initial problem if and only if \(u_0(x)=f_0(x)\) and

$$\begin{aligned} x^p u_{n-1}(qx)=u_n(x)-f_n(x),\qquad n\ge 1. \end{aligned}$$

We can solve this recursively to find that

$$\begin{aligned} u_n(x)=\sum _{j=0}^n q^{pj(j-1)/2} x^{jp} f_{n-j}(q^j x). \end{aligned}$$

These coefficients exhibit the growth

$$\begin{aligned} \sup _{|x|\le r/|q|^n} |u_n(x)|\le CA^n, \end{aligned}$$

for some fixed \(r>0\), due to the restriction on the domain of x. This holds in particular for \(f(x,\epsilon )=f_0(x)\) for which \(u_n(x)=q^{pn(n-1)/2} x^{pn} f_0(q^n x)\). For instance,

$$\begin{aligned} \epsilon x^2\sigma _{q,x}(y)=y-x \end{aligned}$$

has the unique formal solution

$$\begin{aligned} {\hat{y}}(x,\epsilon )=\sum _{n=0}^\infty q^{n(n-1)} x^{2n}(q^nx)\epsilon ^n=\sum _{n=0}^\infty (q^{n^2}\epsilon ^n) x^{2n+1}. \end{aligned}$$

Therefore, the coefficients \(y_k(\epsilon )=y_{2n+1}(\epsilon )=q^{n^2}\epsilon ^n\) grow as

$$\begin{aligned} \sup _{|\epsilon |\le r} |y_{k}(\epsilon )|= r^n|q|^{n^2}=r^{\frac{k-1}{2}}|q|^{(k-1)^2/2\cdot 2}. \end{aligned}$$

This confirms Theorem 1(1) for \(\sigma _q\), showing optimal bounds. In terms of the coefficients of \({\hat{y}}=\sum a_{n,m} x^n \epsilon ^m\), \(a_{2n+1,n}=q^{n^2}\) and \(a_{n,m}=0\) otherwise. Since

$$\begin{aligned} \min \{|q|^{(2n+1)^2/4},|q|^{(2n+1)n}\}=|q|^{(2n+1)^2/4}=|q|^{n^2+n+\frac{1}{4}}, \end{aligned}$$

\({\hat{y}}\) belongs precisely to \({\mathcal {O}}^q_{x^2\epsilon }\).

Example 6

(\(d_q\), \(p= \alpha =1\)) Consider the equation

$$\begin{aligned} \epsilon x^2d_{q}y=y-f_0(x), \end{aligned}$$

where \(f_0(x)=\sum _{n=1}^\infty a_n x^n \in {\mathbb {C}}\{x\}\). Searching for a solution of the form \({\hat{y}}=\sum _{m=0}^{\infty } u_m(x)\epsilon ^{m}\) we see that \(u_m(x)=x^2d_q(u_{m-1})=\cdots =(x^2d_q)^m(f_0)\). Therefore,

$$\begin{aligned} {\hat{y}}=\sum _{n\ge 1,m\ge 0} a_n [n]_q[n+1]_q\cdots [n+m-1]_q x^{n+m} \epsilon ^{m}=\sum _{n\ge 1, m\ge 0} a_{n,m} x^{n+m} \epsilon ^m, \end{aligned}$$

since \((x^2d_q)^m(x^n)=[n]_q[n+1]_q\cdots [n+m-1]_q x^{n+m}\). Recalling (8) we see that \(|a_{n+m,m}|\sim q^{nm}\cdot q^{m(m-1)/2}/(q-1)^{m}\) for \(n,m\rightarrow +\infty \). Therefore, up to a geometric term (depending on q), we see that

$$\begin{aligned} |a_{n+m,m}|\sim q^{nm}\cdot q^{m^2/2}. \end{aligned}$$

On the other hand, Theorem 1(1) provides bounds of type

$$\begin{aligned} \min \{|q|^{(n+m)^2/2}, |q|^{(n+m)m}\}={\left\{ \begin{array}{ll} |q|^{m(n+m)}, & m\le n,\\ |q|^{(n+m)^2/2}, & n\le m. \end{array}\right. } \end{aligned}$$

In particular, if \(n=m\) the theorem asserts bounds of the form \(|q|^{2m^2}\) but the actual term grows as \(|q|^{3m^2/2}\), which is smaller. We will see in the following example that the bounds provided by our results are attained in this case as well.

Example 7

(\(d_q\), \(p= \alpha =1\)) Consider the scalar equation

$$\begin{aligned} \epsilon x^2 d_q(y)=(1+x)y - x\epsilon . \end{aligned}$$
(30)

It has the unique formal power series solution \({\hat{y}}(x,\epsilon )\in {\mathbb {C}}[[x,\epsilon ]]\) given by

$$\begin{aligned} {\hat{y}}(x,\epsilon )=\sum _{n=1}^\infty y_n(\epsilon ) x^n=\sum _{1\le m\le n} a_{n,m} x^n \epsilon ^m,\qquad y_n(\epsilon )=\epsilon \ \prod _{j=1}^{n-1} ([j]_q\epsilon -1). \end{aligned}$$

Therefore, \(a_{n,n}=[n-1]^{!}_q\) and the bounds for the coefficients are attained. Indeed, \(\min \{|q|^{n^2/2},|q|^{nm}\}=|q|^{n^2/2}\) for \(n=m\). On the other hand, the solution is not easy to write when we expand \({\hat{y}}=\sum _{m=1}^\infty u_m(x) \epsilon ^m\). In this case the recurrences are

$$\begin{aligned} u_1(x)=\frac{x}{1+x},\qquad u_m(x)=\frac{x^2}{1+x} d_q(u_{m-1})(x),\quad m\ge 2. \end{aligned}$$

Therefore,

$$\begin{aligned} u_m(x)=\frac{x^m\cdot P_m(x,q)}{(1+x)^mx(1+qx)^{m-1}(1+q^2x)^{m-2}\cdots (1+q^{m-1}x)}, \end{aligned}$$

where \(P_m(x,q)\in {\mathbb {C}}[x,q]\). They can be found using the recursion

$$\begin{aligned} P_{m+1}(x,q)&=\frac{q^m(1+x)^m P_m(qx,q)-(1+qx)(1+q^2x)\cdots (1+q^mx)P_m(x,q)}{q-1}\\&= q^m(1+x)^m x d_qP_m(x,q)+ \frac{q^m(1+x)^m-(-qx;q)_m}{q-1}P_m(x,q). \end{aligned}$$

After some computer calculations, the first values are \(P_2(x,q)=1\), \(P_3(x,q)=-q^{2} x^{2} + q x + q + 1\), and

$$\begin{aligned}&P_4(x,q) = q^{7} x^{5} + x^{4} \left( - 3 q^{6} - 2 q^{5}\right) + x^{3} \left( - 4 q^{6} - 6 q^{5} - 3 q^{4} - 2 q^{3}\right) \\&\quad + x^{2} \left( - q^{6} - 2 q^{5} - q^{4} - q^{3}\right) + x \left( q^{4} + 3 q^{3} + 4 q^{2} + 2 q\right) + + q^{3} + 2 q^{2} + 2 q + 1,\\ P_5(x,q)&= - q^{16} x^{9} - x^{8} \left( - 6 q^{15} - 7 q^{14} - 3 q^{13}\right) \\&\quad - x^{7} \left( - 10 q^{15} - 19 q^{14} - 14 q^{13} - 8 q^{12} - 5 q^{11} - 3 q^{10}\right) \\&\quad - x^{6} \left( - 5 q^{15} - 11 q^{14} - 4 q^{13} + 7 q^{12} + 12 q^{11} + 8 q^{10} + 6 q^{9} + q^{8}\right) \\&\quad - x^{5} \left( - q^{15} - 2 q^{14} + 8 q^{13} + 35 q^{12} + 64 q^{11} + 71 q^{10} + 61 q^{9} + 40 q^{8} + 19 q^{7} + 6 q^{6}\right) \\&\quad - x^{4} \left( 3 q^{13} + 22 q^{12} + 55 q^{11} + 84 q^{10} + 98 q^{9} + 93 q^{8} + 69 q^{7} + 37 q^{6} + 12 q^{5} + 3 q^{4}\right) \\&\quad - x^{3} \left( 4 q^{12} + 15 q^{11} + 30 q^{10} + 44 q^{9} + 54 q^{8} + 50 q^{7} + 34 q^{6} + 18 q^{5} + 8 q^{4} + 2 q^{3}\right) \\&\quad - x^{2} \left( q^{11} + 3 q^{10} + 5 q^{9} + 5 q^{8} - 10 q^{6} - 15 q^{5} - 13 q^{4} - 8 q^{3} - 2 q^{2}\right) \\&\quad - x \left( - q^{8} - 4 q^{7} - 11 q^{6} - 18 q^{5} - 21 q^{4} - 18 q^{3} - 10 q^{2} - 3 q\right) \\&\quad + q^{6} + 3 q^{5} + 5 q^{4} + 6 q^{3} + 5 q^{2} + 3 q +1. \end{aligned}$$

The term \(P_6(x,q)\) has 203 terms, starting with \(x^{14}q^{30}\). It can be shown that

$$\begin{aligned} \text {deg}_x(P_m)=\frac{m(m-1)}{2}-1,\qquad \text {deg}_q(P_m)=\frac{(m-1)(m-2)(m+3)}{6}, \end{aligned}$$

but no easy closed formula seems to be available in the general case.

We conclude this section with examples of Theorem 2.

Example 8

Consider the scalar equation

$$\begin{aligned} \epsilon ^ \alpha d_{q,x}y=y-\frac{x}{1-x}. \end{aligned}$$
(31)

Letting \(\eta =\epsilon ^ \alpha \) and searching for a formal solution of the form \({\hat{y}}_q=\sum _{n=0}^\infty u_n(x)\eta ^n=\sum _{n=0}^\infty v_{ \alpha n}(x)\epsilon ^{ \alpha n}\), we find that \(u_0(x)=\frac{x}{1-x}\), and

$$\begin{aligned} u_{n+1}(x)=d_{q,x}u_n(x),\qquad n\ge 0. \end{aligned}$$

It follows by induction that \(u_n(x)={[n]^{!}_q}/{(x;q)_{n+1}}\), for \(n\ge 1\). Thus, for every \(0<r<1\), \(u_n\in {\mathcal {O}}_b(D_{r/|q|^n})\) and

$$\begin{aligned} \sup _{|x|\le r/|q|^n} |u_n(x)|\le \frac{|[n]^{!}_q|}{(1-r)^{n+1}}. \end{aligned}$$

Therefore, \({\hat{y}}\) has the precise growth described in Theorem 2, since the previous bounds can be written as

$$\begin{aligned} \sup _{|x|\le r/|q|^{ \alpha n/ \alpha }} |v_{ \alpha n}(x)|\le CA^{ \alpha n} |q|^{( \alpha n)^2/2 \alpha ^2}, \end{aligned}$$

for adequate constants \(C,A>0\). Also note that the expansion of \({\hat{y}}_q\) is

$$\begin{aligned} {\hat{y}}_q=\frac{x}{1-x}+\sum _{m=1}^\infty \frac{[m]_q^{!}}{(x;q)_{m+1}} \epsilon ^{ \alpha m}=\frac{x}{1-x}+\sum _{n\ge 0,m\ge 1} \frac{[n+m]_q^{!}}{[n]_q^{!}} x^n \epsilon ^{ \alpha m}. \end{aligned}$$

Thus \(a_{n, \alpha m}=[n+m]_q^{!}/[n]_q^{!}\sim |q|^{(n+m)^2/2}/|q|^{n^2/2}=|q|^{nm+m^2/2} \) as expected.

Remark

Consider (31) with \(f\in {\mathcal {O}}(D_r)\) instead of \(x/(1-x)\in {\mathcal {O}}(D_1)\). We obtain the formal solution and its Borel transform in \(\epsilon \),

$$\begin{aligned} {\hat{y}}(x,\epsilon )=\sum _{n=0}^\infty d_q^n f(x) \epsilon ^n,\qquad {\mathcal {B}}_q({\hat{y}})(\xi ,x):=\sum _{n=0}^\infty \frac{d_q^nf(x)}{[n]_q^{!}} \xi ^n. \end{aligned}$$

In the differential case, i.e., \(q=1\), we see that \({\mathcal {B}}_1({\hat{y}})(\xi ,x)=f(x+\xi )\), \(|x+\xi |<r\) thanks to Taylor’s theorem. However, in the q-difference case, in general only for \(x=0\), we have that \({\mathcal {B}}({\hat{y}})(\xi ,0)=f(\xi )\), \(|\xi |<r\). For instance, if we take the q-exponential \(f(x)=e_q(x):=\sum _{n=0}^\infty x^n/[n]_q^{!}\), we find that

$$\begin{aligned} {\mathcal {B}}_q({\hat{y}})(\xi ,x)=e_q(x)e_q(\xi )\ne e_q(x+\xi ), \end{aligned}$$

if \(x\ne 0, \xi \ne 0\), see [18]. Note we used that \(e_q\) is the unique solution of the problem \(d_qf=f\) and \(f(0)=1\).