Abstract
The aim of this work is to establish the existence, uniqueness and qGevrey character of formal power series solutions of qanalogues of analytic doublysingular equations. Using a new family of norms adapted to qdifferences we find new types of optimal divergence associated with these problems. We also provide some examples to illustrate our results.
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1 Introduction
The genesis of qcalculus 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 qintegral, which is the inverse of Jackson qderivative
In the current literature, it is common to work with the dilation operator
instead. Note that both are related by \(\sigma _q=(q1)x d_q+\text {id}\), where \(\text {id}\) is the identity.
Other sources of qanalogues are Euler’s works on partitions, leading to qexponentials, Gauss’ qbinomial formula and Heine’s qhypergeometric series with applications in Number Theory, see [18]. These are examples of functions y satisfying qdifference equations, i.e., relations of the form
Writing this equation in terms of \(d_q\) and noticing that \(d_qf\rightarrow f'\) as \(q\rightarrow 1\), we find that qdifference equations are a discrete counterpart to differential equations. For historical accounts of qcalculus and qdifference equations we refer to the survey [15].
In the analytic setting, linear algebraic qdifference 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 qdifference framework when \(q>1\), the divergence is given by the factor \(q^{sn^2/2}, s>0\). These series are referred as qGevrey. 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 qsummability have been developed, taking into account different notions of asymptotic expansions on usual sectors or qspirals. They adapt the use Borel and Laplace transformations by suitable qanalogues. For instance, using the Jacobi’s theta function [25, 30], the two qanalogues 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 doublysingular systems of analytic ODEs of the form
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 qanalogues 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 qdifference 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
with similar hypotheses as before. Here we use the notation \(\sigma _{q,x}(y)(x,\epsilon ):=y(qx,\epsilon )\) and \((q1)x d_{q,x}+\text {id}=\sigma _{q,x}\) 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 qsummability [14].
For each equation we will first establish the existence and uniqueness of a solution
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 qNagumo norms. Incidentally, we can also treat by the same technique both equations in the Fuchsianlike 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
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.
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.
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 qdifference 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
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 \(n\ge 0\),
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 qdifference 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 qanalogues, \(d_q\) and \(\sigma _q\), while Sect. 3 includes some spaces of qGevrey series. In Sect. 4 we develop qNagumo 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 nonmodified qNagumo 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 nonnegative 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 qanalogues, including qfactorials and qGevrey series. From now on we fix \(q\in {\mathbb {C}}\) with \(q>1\). We write
for the qanalogue to \(\lambda \), where the principal branch of the logarithm is considered for \(\lambda >0\), and which reduces to
We have that \([0]_q=0\) and
For future use, we note that \([n]_q\le [n]_{q}\), for \(n\in {\mathbb {N}}\), and
These constants appear naturally while considering Jackson’s qderivative (1) of a function f. As before, \(\sigma _q(f)(x):=f(qx)\). For analytic functions \(f\in {\mathcal {O}}(D_r)\),
They can be computed term by term using power series expansion and the rules
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
We recall the coefficients
The second one is convergent as we can compare it with a geometric series. The qfactorial is defined accordingly as
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,
Thus
Another useful qanalogue is the qbinomial coefficient
which is a polynomial in q of degree \(j(nj)\) satisfying qPascal’s formula
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.,
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 logconvex, i.e., when it holds that
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, \(M_{n}/M_{n1}=[n]_{q}^s\) which is increasing in n.
In general, qfactorials determine the divergence rate of solutions of singular qdifference equations. Two classical examples are the following.
Example 1
A qanalogue to Euler’s equation is
which is divergent for \(q>1\). In contrast, the qdifference equation
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 \(\lim _{q\rightarrow 1^+} {\hat{Y}}_q(x)=\frac{x}{1x}\) 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 sqGevrey, where \(s\ge 0\) and \(q>1\), if there are constants \(C=C(q),A=A(q)>0\) such that
We will denote the space of such series by \({\mathbb {C}}[[x]]_{q,s}\).
Note that the qfactorial is desirable when working with qdifference 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 sqGevrey is equivalent to request that
for some constants \(B,D>0\) that might also depend on q. Indeed, since
we have \([n]_{q}^{!}=(q^{1};q^{1})_{n}\frac{q^{n(n+1)/2}}{(q1)^n}\), which entails the limit
This means that \([n]_{q}^{!}\) is asymptotically equivalent to \( c(q)(q1)^{n}q^{n(n+1)/2}\), c.f., [24, p. 55]. Thus, \([n]_{q}^{!}\) is asymptotically equivalent to \( c(q)(q1)^{n} q^{n(n+1)/2}\), and \([n_q^{!}\) is to \( c(q) q1^{n} q^{n(n+1)/2}\). We also note the simple bound
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.
\({\hat{f}}\) is sqGevrey if and only if it is 1\(q^s\)Gevrey.

2.
Fix \(p\in {\mathbb {N}}^+\) and write \({\hat{f}}(x)=\sum _{j=0}^{p1} x^j {\hat{f}}_j(x^p)\), where \({\hat{f}}_j(t)=\sum _{n=0}^\infty a_{np+j} t^n\). Then \({\hat{f}}(x)\) is sqGevrey (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 \(C_j=CA^jq^{sj^2/2}\), \(A_j=A^p q^{pjs}\). The converse is similar.
3 Some spaces of qGevrey series
This section introduces two spaces of qGevrey 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
for \(y_n\in {\mathbb {C}}[[x_2]]\) and \(u_m\in {\mathbb {C}}[[x_1]]\). First, we consider the space
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.
\({\hat{f}}\in {\mathcal {O}}_{0}^q\),

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.
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
and analogously for \(u_m(x_1)\). Now, if (2) holds, by Cauchy’s inequalities,
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
confluent to \(\left[ (1x_1)(1x_2)\right] ^{1}\) 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
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 \(U_{q,1}=\{x_1\in {\mathbb {C}}: 1x_1>1\}\) 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
corresponding to equation (5) in Theorem 1(2).
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 1qGevrey in \(x_1^px_2^ \alpha \) if there are constants \(C=C(q),A=A(q)>0\) such that
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.
\({\hat{f}}\in {\mathcal {O}}^q_{x_1^p x_2^ \alpha }\),

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
$$\begin{aligned} \sup _{x_2\le r} y_n(x_2)\le DB^nq^{n^2/2p},\qquad \sup _{x_1\le r/q^{n/ \alpha }} u_n(x_1)\le DB^n. \end{aligned}$$
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.
\({\mathcal {O}}^{q}_{0}\) 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.
\({\mathcal {O}}^q_{x_1^p x_2^ \alpha }\subset {\mathcal {O}}^{q^{1/ \alpha }}_{0}\).

3.
\({\hat{f}}=\sum _{j=0}^{p1} {\hat{f}}_j(x_1^p,x_2) x_1^j\in {\mathcal {O}}_{0}^q\) if and only if \({\hat{f}}_j(z,x_2)\in {\mathcal {O}}_{0}^{q^{p}}\), for \(j=0,1,\dots ,p1\).

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 \(j=0,1,\dots ,p1\), \(l=0,1,\dots , \alpha 1\).
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
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 qNagumo 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
Choosing \(\rho \) adequately, the following assertions hold, see Fig. 2.
Lemma 4
If \(\rho =r/q\), then

1.
\(d_n(t)d_n(s)\le q^nts\).

2.
\(d_n(qt)=d_{n+1}(t)\), for \(t\le r/q^{n+1}\).

3.
\(d_{n+1}(t)\le d_n(t)\le r(11/q)\), 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 qNagumo norm by
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
This family of norms takes into account that factor \(rq^nx\) 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.
\(\Vert f+g\Vert _n\le \Vert f\Vert _n+\Vert g\Vert _n\) and \(\Vert fg\Vert _{n+m}\le \Vert f\Vert _n \Vert g\Vert _m\).

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

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
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
thanks to Lemma 4(2) and (3). Thus, we find that
If \(x\le \rho /q^{n+1}\), using the Maximum Modulus Principle and (17) we see that
Therefore,
Since \(q1\le q1\), these inequalities establish (2).
Finally, to prove (3) simply note that by definition of \(\Vert f\Vert _n\) we have
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
The previous lemma shows that
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 qNagumo 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.
Apply rank reduction in \(\epsilon \) to assume \( \alpha =1\).

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.
Using the supremum norm in \(D_r\) or \(D_{r/q^n}\) (\({\hat{y}}\) as power series in x) or the qNagumo 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 logconvex sequence \(\{M_n\}_{n\ge 1}\) to find a sequence of inequalities for \(z_n/M_n\).

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
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
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 lowertriangular 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
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
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
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
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 \(y\mapsto yy_0(\epsilon )\) in the initial equation (19), we can assume now that \(y_0(\epsilon )=0\). Thus we obtain a similar qdifference 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
for the first system. The same recurrence holds for the second system with \([np]_{q}\) replaced by \(q^{np}\). 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 \(0\le m\le nk\), 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 \(n_{l,j} < nm\le n\), thus no component of \({\hat{y}}_n\) appears in the former sum. Finally, the leftside 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 qNagumo 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
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
which is logconvex, see Remark at page 6 and (14). Recalling (12) we find that
For the second system we use the corresponding inequality to (20), to obtain that \(z_n=\Vert y_n\Vert \) satisfies (21) with \([np]_{q}\) replaced by \(q^{np}\). Then we divide by
and noticing that \(q^{np}/M_n\le 1/M_{np}\), we arrive again to (22).
Step 4: The majorized problem. Define recursively \(w_n\) by \(w_1=\Vert y_1\Vert \) and
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
where
define analytic (in this case, actually entire) functions of \(\tau \). Consider the map
which is analytic at \((\tau ,w)=(0,0)\), due to estimates of the form
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
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
In this case, the reduction on the radius on \(\epsilon \) comes from \(\left( \epsilon [n]_q I_NA_{0*}(\epsilon )\right) ^{1}\). 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
and using the inequality \(\Vert (IB)^{1}\Vert \le \frac{1}{1\Vert B\Vert }\), valid for \(\Vert B\Vert <1\) (Neumann series), for any matrix norm, we find that
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
It follows that
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 \(\left( \epsilon q^n I_NA_{0*}(\epsilon )\right) ^{1}\) 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 \(y\mapsto yu_0(x)\) 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 qdifference equations
where the sum \((*_k)\) has the same structure as before. For the second system we obtain (28) with \(x^{p}\sigma _q(u_{n1})\) instead of \(x^{p+1}d_q(u_{n1})\).
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 qNagumo norms developed in Sect. 4 establish that
In fact, the qderivative \(d_q\) is controlled by the inequality
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
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
where \(L=Bq/r(q1)\) 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
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
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
In this case we choose
which is logconvex since \(M_n/M_{n1}=q^{n+\frac{1}{2}}\) is increasing in n, and satisfies \(q^n/M_n\le 1/M_{n1}\). Therefore, dividing the previous inequality by \(M_n\) and proceeding as before, we find that
for some \(B,D>0\). Recalling the definition of qNagumo norms, this means that
for \(x\le \rho /q^{n}=r/q^{n+1}\), where \(L=Bq^2/r(q1)>0\).
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
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
Computing \(d_{q,x}(y)\), it follows using (10) and (7) that
Then \(w(z,\eta )=(y_{l,j})_{0\le l<p, 0\le j< \alpha }\), written in lexicographical order, satisfies
where \(DG_w(0,0,0)=[p]_q^{1}D_q{\widetilde{A}}\), \(D_q=\text {diag}(I_{ \alpha N}, q^{1} I_{ \alpha N},\dots ,q^{(p1)} I_{ \alpha N})\), and \({\widetilde{A}}\) is a block lowertriangular 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
Therefore, this reduction would force to change the nature of the problem by introducing q on the nonqdifference 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 \(w=ya\), 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
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
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
where \( \alpha \in {\mathbb {N}}^+\), \(y\in {\mathbb {C}}^N\) and A(0) is an invertible matrix. It follows that
is the unique formal power series solution of the problem. Although \(F(x,\epsilon ,y)=A(x)yb(x)\) only satisfies that \(F(0,0,0)=0\) when \(b(0)=0\), we can make the change of variables \(w=y{\hat{y}}_q(0,0)=yA(0)^{1}b(0)\) to obtain the equivalent system
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
generating the solution
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 \({\hat{y}}_1=(1x\epsilon ^ \alpha )^{1}\) 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
If \( \alpha =1\) and \(b(x)=(1x)^{1}\), we recover the series \(M(x,\epsilon )\) of Example 2.
Example 4
(\(d_q\), case \(p=0\)) Consider the problem
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 \(w=ya_0\). Solving directly for \({\hat{y}}=\sum _{m=0}^{\infty } u_m(x)\epsilon ^{ \alpha m}\) we see that \(u_m(x)=xd_q(u_{m1})=\cdots =(xd_q)^m(f_0)\). Therefore,
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
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
We can solve this recursively to find that
These coefficients exhibit the growth
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 \(u_n(x)=q^{pn(n1)/2} x^{pn} f_0(q^n x)\). For instance,
has the unique formal solution
Therefore, the coefficients \(y_k(\epsilon )=y_{2n+1}(\epsilon )=q^{n^2}\epsilon ^n\) grow as
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
\({\hat{y}}\) belongs precisely to \({\mathcal {O}}^q_{x^2\epsilon }\).
Example 6
(\(d_q\), \(p= \alpha =1\)) Consider the equation
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 \(u_m(x)=x^2d_q(u_{m1})=\cdots =(x^2d_q)^m(f_0)\). Therefore,
since \((x^2d_q)^m(x^n)=[n]_q[n+1]_q\cdots [n+m1]_q x^{n+m}\). Recalling (8) we see that \(a_{n+m,m}\sim q^{nm}\cdot q^{m(m1)/2}/(q1)^{m}\) for \(n,m\rightarrow +\infty \). Therefore, up to a geometric term (depending on q), we see that
On the other hand, Theorem 1(1) provides bounds of type
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
It has the unique formal power series solution \({\hat{y}}(x,\epsilon )\in {\mathbb {C}}[[x,\epsilon ]]\) given by
Therefore, \(a_{n,n}=[n1]^{!}_q\) 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
Therefore,
where \(P_m(x,q)\in {\mathbb {C}}[x,q]\). They can be found using the recursion
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
The term \(P_6(x,q)\) has 203 terms, starting with \(x^{14}q^{30}\). It can be shown that
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
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 \(u_0(x)=\frac{x}{1x}\), and
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
Therefore, \({\hat{y}}\) has the precise growth described in Theorem 2, since the previous bounds can be written as
for adequate constants \(C,A>0\). Also note that the expansion of \({\hat{y}}_q\) is
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 \(x/(1x)\in {\mathcal {O}}(D_1)\). We obtain the formal solution and its Borel transform in \(\epsilon \),
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 qdifference 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 qexponential \(f(x)=e_q(x):=\sum _{n=0}^\infty x^n/[n]_q^{!}\), we find that
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 \(f(0)=1\).
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The authors express their gratitude to the referees for their careful review and valuable suggestions that helped to improve the manuscript. The authors appreciate the suggestions inspiring the remark at the beginning of the section of examples.
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Research supported by the project PID2022139631NBI00 of Ministerio de Ciencia e Innovación, Spain. The first author is partially supported by Universidad Nacional de Colombia under the Proyect 56664(2023). The first author acknowledge Universidad de Alcalá (UAH) for its hospitality and support under the “Giner de los Rios” program (2021) where this work started. The second author is partially supported by Ministerio de Ciencia e InnovaciónAgencia Estatal de Investigación MCIN/AEI/10.13039/501100011033 and the European Union “NextGenerationEU”/PRTR, under grant TED2021129813AI00.
Appendix: qNagumo norms suitable for confluence
Appendix: qNagumo norms suitable for confluence
Here we include nonmodified qNagumo norms suitable for confluence, useful to recover the known results on the rate of divergence of solutions of (2).
Let \(q>1\) and \(f\in {\mathcal {O}}_b(D_r)\). The nth qNagumo norm is defined by
Note that if \(\Vert f\Vert '_n\) is finite, we have that
for \(x\le \rho /q^n<r/q^n\). For these norms we have the following properties.
Lemma 6
Let \(q>1\), \(n,m\in {\mathbb {N}}\) and \(f,g \in {\mathcal {O}}_b(D_r)\). Then:

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.
\(\Vert d_q(f)\Vert '_{n+1}\le eq^n(n+1)\Vert f\Vert '_n\).

3.
\(\Vert \sigma _q(f)\Vert '_{n+1}\le r\Vert f\Vert '_n\).
Proof
For (1) assume \(n\le m\). Since \(rq^{n+m}x\le rq^mx\le rq^nx\) and \(r/q^{n+m}\le r/q^m\le r/q^n\), we find
for all \(x\le r/q^{n+m}\). This proves the inequality for the product.
For (2) fix x such that \(x< r/q^{n+1}\) and choose \(\rho >0\) with \(qx+\rho <r/q^n\). Notice that if \(1\le t\le q\), then \(tx+\rho <r/q^n\). Therefore, the disk \(D(tx,\rho )\) is contained in \(D(0,r/q^n)\). By Cauchy’s formulas we can write
where \(\gamma \) is the boundary of \(D(tx,\rho )\). It follows that
Therefore,
Choosing \(q^n\rho =\frac{rq^{n+1}x}{n+1}\) we find that
and the result follows from the wellknown inequality \((1\frac{1}{n+1})^{n}<e\).
To prove (3) simply note that by definition of \(\Vert f\Vert '_n\) we have
Therefore, \(f(qx)(rq^{n+1}x)^{n+1}\le \Vert f\Vert '_n (rq^{n+1}x)<r\Vert f\Vert '_n\), as needed. \(\square \)
Corollary 1
(Proposition 5.1 [7]) Consider the system of doublysingular equations (2) with the same hypotheses on F as in Theorem 1. Then the problem has a unique formal power series solution \({\hat{y}}=\sum _{n,m\ge 0} a_{n,m} x^n \epsilon ^m\in {\mathbb {C}}[[x,\epsilon ]]^N\) which is 1Gevrey in the monomial \(x^p\epsilon ^\sigma \), i.e., there are constants \(C,A>0\) such that
Proof
For each \(q>1\) consider (4) for \( \alpha =1\). By Theorem 1 there is a unique formal power series solution \({\hat{y}}_q=\sum _{n=0}^\infty y_{n}^{[q]}(\epsilon ) x^n=\sum _{m=0}^\infty u_{m}^{[q]}(x) \epsilon ^m=\sum _{n,m\ge 0} a_{n,m}(q) x^n\epsilon ^m \in {\mathbb {C}}[[x,\epsilon ]]^N\). Letting \(q\rightarrow 1^+\), it is easy to check that \({\hat{y}}=\lim _{q\rightarrow 1^+} {\hat{y}}_q\) (term by term) as in (3) satisfies (2).
A second look at the proof of Theorem 1, Steps 3 and 4, shows that if we modify \({\widetilde{b}}, {\widetilde{A}}\), and each \({\widetilde{A}}_I\) changing \(M_n=([n]!_{q})^{1/p}\) by 1, the same argument works to conclude that
for constants \(C,L>0\) independent of q. Therefore, letting \(q\rightarrow 1^+\), we find that
To obtain the bounds as a power series in \(\epsilon \), we repeat the corresponding proof but now using \(z_n=\Vert u_n^{[q]}\Vert _n'\). In this case, the inequalities (29) take the form
Thus, after dividing by \(M_n=n!\) and arguing as usual, we conclude that \(\Vert u_n^{[q]}\Vert _n\le DB^n n!\), for some constants \(D,B>0\) independent of q. Therefore,
Letting \(q\rightarrow 1^+\), we find that \(\sup _{x\le \rho } u_n(x)\le \frac{DB^n}{(r\rho )^n} n!\), as required. Undoing the rank reduction by using the bounds
we obtain the general case. Finally, Cauchy’s inequalities lead to (32). \(\square \)
Remark
If \(n= \alpha k+j\), \(0\le j< \alpha \), it follows that
Letting \(q\rightarrow 1^+\) we recover the inequality \(k!\le ( \alpha k)!^{1/ \alpha }\le n!^{1/ \alpha }\) employed above.
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Carrillo, S.A., Lastra, A. qNagumo norms and formal solutions to singularly perturbed qdifference equations. Math. Ann. (2024). https://doi.org/10.1007/s00208024030073
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DOI: https://doi.org/10.1007/s00208024030073