A note on $G_q$-summability of formal solutions of some linear $q$-difference-differential equations

TL;DR

This paper investigates the summability properties of formal solutions to certain linear q-difference-differential equations, enhancing previous results by providing improved conditions or methods for q-G-summability.

Contribution

It offers new insights or methods that extend the understanding of q-G-summability for formal solutions of linear q-difference-differential equations.

Findings

Improved conditions for q-G-summability.

Extended the class of equations with summable solutions.

Refined previous results by Tahara-Yamazawa.

Abstract

The paper discusses the summability of formal solutions of some linear q-difference-differential equations, and improves the previous result in [Tahara-Yamazawa, Opsucula Math. 35 (2015), 713-738].

Equations

$$\sum _{j+\delta |\alpha |\le m}{a}_{j,\alpha }(t,z)({\sigma }_q{)}^j{\partial }_z^{\alpha }X=F(t,z)$$

$$X(t,z)=\sum _{n\ge 0}{X}_n(z)t^n\in {\mathcal{O}}_R[[t]]$$

$${\mathcal{Z}}_{\lambda }=\{-\lambda q^m\in \mathbb{C};m\in \mathbb{Z}\},$$

$${\mathcal{Z}}_{\lambda ,ϵ}=\bigcup _{m\in \mathbb{Z}}\{t\in \mathbb{C}\setminus \{0\};|1+\lambda q^m/t|\le ϵ\}.$$

$$\biggl{|}W(t,z)-\sum_{n=0}^{N-1}X_{n}(z)t^{n}\biggr{|}\leq\frac{MH^{N}}{\epsilon}q^{N(N-1)/2}|t|^{N}$$

$$\mathrm{ord}_{t}(f)=\min\{k\in\mathbb{N}\,;\,(\partial_{t}^{k}f)(0,z)\not\equiv 0\enskip\mbox{near $z=0$}\}$$

$$N_t(\text{ref}1.1)=\text{mbox}theconvexhullof\bigcup _{j+\delta |\alpha |\le m}C(j,{\mathrm{ord}}_t(a_{j,\alpha })).$$

$$N_t(\text{ref}1.1)=\{(x,y)\in {\mathbb{R}}^2;x\le m,y\ge \max \{0,x-m_0\}\}.$$

$$|\alpha |>0⟹(j,{\mathrm{ord}}_t(a_{j,\alpha }))\in int(N_t(\text{ref}1.1)),$$

$$\mathrm{ord}_{t}(a_{j,\alpha})\geq\left\{\begin{array}[]{ll}\max\{0,j-m_{0}\},&\mbox{if $|\alpha|=0$},\\[2.0pt] \max\{1,j-m_{0}+1\},&\mbox{if $|\alpha|>0$}.\end{array}\right.$$

$$a_{j,0}(t,z)=t^{j-m_{0}}b_{j,0}(t,z)\quad\mbox{for $m_{0}<j\leq m$}$$

$$a_{m_0,0}(0,0)\ne 0\text{mbox}and{b}_{m,0}(0,0)\ne 0.$$

$$P(\tau ,z)=\sum _{m_0<j\le m}\frac{b_{j,0}(0,z)}{q^{j(j-1)/2}}{\tau }^{j-m_0}+\frac{a_{m_0,0}(0,z)}{q^{m_0(m_0-1)/2}}$$

$$S=\bigcup _{i=1}^{m-m_0}\{t={\tau }_i\eta ;\eta >0\}.$$

$$\mathrm{ord}_{t}(a_{j,\alpha})\geq j-m_{0}+2,\quad\mbox{if $|\alpha|>0$ and $m_{0}\leq j<m$}$$

$$u(\xi ,z)=\sum _{k\ge 0}\frac{X_k(z)}{q^{k(k-1)/2}}{\xi }^k$$

$$|u^{*}(\lambda q^{m},z)|\leq CH^{m}q^{m^{2}/2}\quad\mbox{on $D_{R_{1}}$},\quad m=0,1,2,\ldots.$$

$$|a_n|\le \frac{A{H}^n}{q^{n(n-1)/2}},n=0,1,2,\dots .$$

$$|f(t)|\leq M\exp\Bigl{(}\dfrac{(\log|t|)^{2}}{2\log q}+\alpha\log|t|\Bigr{)}\quad\mbox{on $\mathbb{C}\setminus\{0\}$}$$

$$\sum _{j+\delta |\alpha |\le m}{A}_{j,\alpha }(t,z)({\sigma }_{q_1}{)}^{2j}{\partial }_z^{\alpha }Y=G(t,z)$$

$$A_{j,\alpha }(t,z)=a_{j,\alpha }(t^2,z)(j+\delta |\alpha |\le m),$$

$$Y(t,z)=X(t^2,z)=\sum _{k\ge 0}{X}_k(z)t^{2k},$$

$$G(t,z)=F(t^2,z).$$

$$N_t(\text{ref}4.1)=\{(x,y)\in {\mathbb{R}}^2;x\le 2m,y\ge \max \{0,x-2m_0\}\}$$

$$\mathrm{ord}_{t}(A_{j,\alpha})\geq\left\{\begin{array}[]{ll}\max\{0,2j-2m_{0}\},&\mbox{if $|\alpha|=0$},\\[2.0pt] \max\{2,2j-2m_{0}+2\},&\mbox{if $|\alpha|>0$}.\end{array}\right.$$

$$A_{j,0}(t,z)=t^{2j-2m_{0}}B_{j,0}(t,z)\quad\mbox{for $m_{0}<j\leq m$}$$

$$P_1(\rho ,z)$$

$$=\sum _{m_0<j\le m}\frac{b_{j,0}(0,z)}{{q_1}^{2j(2j-1)/2}}{\rho }^{2j-2m_0}.+\frac{a_{m_0,0}(0,z)}{{q_1}^{2m_0(2m_0-1)/2}}.$$

$$S_1=\bigcup _{i=1}^{2m-2m_0}\{t={\rho }_i\eta ;\eta >0\}.$$

$$u_1(\xi ,z)=\sum _{k\ge 0}\frac{X_k(z)}{{q_1}^{2k(2k-1)/2}}{\xi }^{2k}.$$

$$u_1(\xi ,z)=u(q^{-1/4}{\xi }^2,z),$$

$$P_1(\lambda ,z)=q^{-m_0/4}P(q^{-1/4}{\lambda }^2,z),$$

$$u^{*}_{1}(\xi,z)=\sum_{n\geq 2\mu}w_{n}(\xi,z)+\sum_{0\leq k<\mu}\frac{X_{k}(z)}{{q_{1}}^{2k(2k-1)}}\xi^{2k}\quad\mbox{on $S_{\theta_{1}}(\rho)\times D_{R_{1}}$}$$

Full text

Key words and phrases: $q$-difference-differential equations, summability, formal power series solutions, $q$-Gevrey asymptotic expansions.

2010 Mathematics Subject Classification Numbers: Primary 35C20; Secondary 35A01, 39A13.

000 The first author is supported by JSPS KAKENHI Grant Number JP15K04966.

1 Introduction

Let $(t,z)$ be the variable in $\mathbb{C}_{t}\times\mathbb{C}_{z}^{d}$. Let $q>1$. For a function $f(t,z)$ we define a $q$-shift operator $\sigma_{q}$ in $t$ by $\sigma_{q}(f)(t,z)=f(qt,z)$.

In this note, we consider a linear $q$-difference-differential equation

[TABLE]

under the following assumptions:

(1) $q>1$, $\delta>0$ and $m\in\mathbb{N}^{*}(=\{1,2,\ldots\})$;

(2) $a_{j,\alpha}(t,z)$ ($j+\delta|\alpha|\leq m$) and $F(t,z)$ are holomorphic functions in a neighborhood of $(0,0)\in\mathbb{C}_{t}\times\mathbb{C}_{z}^{d}$;

(3) (1.1) has a formal power series solution

[TABLE]

where ${\mathcal{O}}_{R}$ denotes the set of all holomorphic functions on $D_{R}=\{z\in\mathbb{C}^{d}\,;\,|z_{i}|<R\enskip(i=1,\ldots,d)\}$.

Our basic problem is:

Problem 1.1. Under what condition can we get a true solution $W(t,z)$ of (1.1) which admits $\hat{X}(t,z)$ as a $q$-Gevrey asymptotic expansion of order 1 (in the sense of Definition 1.2 given below) ?

For $\lambda\in\mathbb{C}\setminus\{0\}$ and $\epsilon>0$ we set

[TABLE]

It is easy to see that if $\epsilon>0$ is sufficiently small the set ${\mathscr{Z}}_{\lambda,\epsilon}$ is a disjoint union of closed disks. For $r>0$ we write $D_{r}^{*}=\{t\in\mathbb{C}\,;\,0<|t|<r\}$. The following definition is due to Ramis-Zhang [8].

Definition 1.2. (1) Let $\hat{X}(t,z)=\sum_{n\geq 0}X_{n}(z)t^{n}\in{\mathcal{O}}_{R}[[t]]$ and let $W(t,z)$ be a holomorphic function on $(D_{r}^{*}\setminus{\mathscr{Z}}_{\lambda})\times D_{R}$ for some $r>0$. We say that $W(t,z)$* admits $\hat{X}(t,z)$ as a $q$-Gevrey asymptitoc expansion of order 1*, if there are $M>0$ and $H>0$ such that

[TABLE]

holds on $(D_{r}^{*}\setminus{\mathscr{Z}}_{\lambda,\epsilon})\times D_{R}$ for any $N=0,1,2,\ldots$ and any sufficiently small $\epsilon>0$.

(2) If there is a $W(t,z)$ as above, we say that the formal solution $\hat{X}(t,z)$ is $G_{q}$-summable in the direction $\lambda$.

A partial answer to Problem 1.1 was given in Tahara-Yamazawa [11]: in this paper, we will give an improvement of the result in [11]. As in [11], we will use the framework of $q$-Laplace and $q$-Borel transforms via Jacobi theta function, developped by Ramis-Zhang [8] and Zhang [10].

Similar problems are discussed by Zhang [9], Marotte-Zhang [5] and Ramis-Sauloy-Zhang [7] in the $q$-difference equations, and by Malek [3, 4], Lastra-Malek [1] and Lastra-Malek-Sanz [2] in the case of $q$-difference-differential equations. But, their equations are different from ours.

2 Main results

For a holomorphic function $f(t,z)$ in a neighborhood of $(0,0)\in\mathbb{C}_{t}\times\mathbb{C}_{z}^{d}$, we define the order of the zeros of the function $f(t,z)$ at $t=0$ (we denote this by $\mathrm{ord}_{t}(f)$) by

[TABLE]

where $\mathbb{N}=\{0,1,2,\ldots\}$.

For $(a,b)\in\mathbb{R}^{2}$ we set $C(a,b)=\{(x,y)\in\mathbb{R}^{2}\,;\,x\leq a,y\geq b\}$. We define the $t$-Newton polygon $N_{t}(\ref{1.1})$ of equation (1.1) by

[TABLE]

In this note, we will consider the equation (1.1) under the following conditions (A1) and (A2):

(A1) There is an integer $m_{0}$ such that $0\leq m_{0}<m$ and

[TABLE]

(A2) Moreover, we have

[TABLE]

where $int(N_{t}(\ref{1.1}))$ denotes the interior of the set $N_{t}(\ref{1.1})$ in $\mathbb{R}^{2}$.

The figure of $N_{t}(\ref{1.1})$ is as in Figure 1. In Figure 1, the boundary of $N_{t}(\ref{1.1})$ consists of a horizontal half-line $\Gamma_{0}$, a segment $\Gamma_{1}$ and a vertical half-line $\Gamma_{2}$, and $k_{i}$ is the slope of $\Gamma_{i}$ for $i=0,1,2$.

Lemma 2.1**.**

If (A1) and (A2) are satisfied, we have

[TABLE]

By the condition (2.1), we have the expression

[TABLE]

for some holomorphic functions $b_{j,0}(t,z)$ ($m_{0}<j\leq m$) in a neighborhood of $(0,0)\in\mathbb{C}\times\mathbb{C}_{z}^{d}$. We suppose:

[TABLE]

We set

[TABLE]

and denote by $\tau_{1},\ldots,\tau_{m-m_{0}}$ the roots of $P(\tau,0)=0$. By (2.3) we have $\tau_{i}\neq 0$ for all $i=1,2,\ldots,m-m_{0}$. The set $S$ of singular directions at $z=0$ is defined by

[TABLE]

In [11], we have shown the following result.

Theorem 2.2** (Theorem 2.3 in [11]).**

(1)* Suppose the conditions (A1), (A2) and (2.3). Then, if equation (1.1) has a formal solution $\hat{X}(t,z)=\sum_{n\geq 0}X_{n}(z)t^{n}\in{\mathcal{O}}_{R}[[t]]$, we can find $A>0$, $h>0$ and $0<R_{1}<R$ such that $|X_{n}(z)|\leq Ah^{n}q^{n(n-1)/2}$ on $D_{R_{1}}$ for any $n=0,1,2,\ldots$.*

(2)* In addition, if the condition*

[TABLE]

is satisfied, for any $\lambda\in\mathbb{C}\setminus(\{0\}\cup S)$ the formal solution $\hat{X}(t,z)$ is $G_{q}$-summable in the direction $\lambda$. In other words, there are $r>0$, $R_{1}>0$ and a holomorphic solution $W(t,z)$ of (1.1) on $(D_{r}^{*}\setminus{\mathscr{Z}}_{\lambda})\times D_{R_{1}}$ such that $W(t,z)$ admits $\hat{X}(t,z)$ as a $q$-Gevrey asymptitoc expansion of order 1.

In this paper, we remove the additional condition (2.5) from the part (2) of Theorem 2.2. We have

Theorem 2.3**.**

*Suppose the conditions (A1), (A2) and (2.3). Then, for any $\lambda\in\mathbb{C}\setminus(\{0\}\cup S)$ the formal solution $\hat{X}(t,z)$ *(in (1.2)) is $G_{q}$-summable in the direction $\lambda$.

To prove this, we use the framework of $q$-Laplace and $q$-Borel transforms developped by Rramis-Zhang [8]. By (1) of Theorem 2.2 we know that the formal $q$-Borel transform of $\hat{X}(t,z)$ in $t$

[TABLE]

is convergent in a neighborhood of $(0,0)\in\mathbb{C}_{\xi}\times\mathbb{C}_{z}^{n}$. For $\lambda\in\mathbb{C}\setminus\{0\}$ and $\theta>0$ we write $S_{\theta}(\lambda)=\{\xi\in\mathbb{C}\setminus\{0\}\,;\,|\arg\xi-\arg\lambda|<\theta\}$. Then, to show Theorem 2.3 it is enough to prove the following result.

Proposition 2.4**.**

For any $\lambda\in\mathbb{C}\setminus(\{0\}\cup S)$ there are $\theta>0$, $R_{1}>0$, $C>0$ and $H>0$ such that $u(\xi,z)$ has an analytic extension $u^{*}(\xi,z)$ to the domain $S_{\theta}(\lambda)\times D_{R_{1}}$ satisfying the following condition:

[TABLE]

3 Some lemmas

Before the proof of Proposition 2.4, let us give some lemmas which are needed in the proof of Proposition 2.4.

The following is the key lemma of the proof of Proposition 2.4.

Lemma 3.1**.**

Let $q>1$. Let $f(t,z)$ be a function in $(t,z)$.

(1)* We have $\sigma_{q}(f)(t,z)=(\sigma_{\sqrt{q}})^{2}(f)(t,z)$.*

(2)* We set $F(t,z)=f(t^{2},z)$: then we have $\sigma_{q}(f)(t^{2},z)=\sigma_{\sqrt{q}}(F)(t,z)$. Similarly, we have $(\sigma_{q})^{m}(f)(t^{2},z)=(\sigma_{\sqrt{q}})^{m}(F)(t,z)$ for any $m=1,2,\ldots$.*

Proof.

(1) is clear. (2) is verified as follows: $\sigma_{q}(f)(t^{2},z)=f(qt^{2},z)=f((\sqrt{q}t)^{2},z)$ $=F(\sqrt{q}t,z)=\sigma_{\sqrt{q}}(F)(t,z)$. The equality $(\sigma_{q})^{m}(f)(t^{2},z)=(\sigma_{\sqrt{q}})^{m}(F)(t,z)$ can be proved in the same way. ∎

The following result is proved in [Proposition 2.1 in [6]]:

Proposition 3.2**.**

Let $\hat{f}(t)=\sum_{n\geq 0}a_{n}t^{n}\in\mathbb{C}[[t]]$. The following two conditions are equivalent:

(1)* There are $A>0$ and $H>0$ such that*

[TABLE]

(2)* $\hat{f}(t)$ is the Taylor expansion at $t=0$ of an entire function $f(t)$ satisfying the estimate*

[TABLE]

for some $M>0$ and $\alpha\in\mathbb{R}$.

4 Proof of Proposition 2.4

We set $q_{1}=q^{1/4}$, replace $t$ by $t^{2}$ in (1.1), and apply Lemma 3.1 to the equation (1.1): then (1.1) is rewritten into the form

[TABLE]

where

[TABLE]

We can regards (4.1) as a $q_{1}$-difference-differential equation, and in this case, the order of the equation is $2m$ in $t$. Therefore, the $t$-Newton polygon $N_{t}(\ref{4.1})$ of (4.1) (as a $q_{1}$-difference equation) is

[TABLE]

which is as in Figure 2.

Moreover, we have

[TABLE]

By (2.2) we have

[TABLE]

for $B_{j,0}(t,z)=b_{j,0}(t^{2},z)$ ($m_{0}<j\leq m$). The set $S_{1}$ of singular directions of (4.1) is defined by using

[TABLE]

Let $\rho_{1},\ldots,\rho_{2m-2m_{0}}$ be the roots of $P_{1}(\rho,0)=0$: then $S_{1}$ is defined by

[TABLE]

Let $u_{1}(\xi,x)$ be the $q_{1}$-formal Borel transform of $Y(t,x)$, that is,

[TABLE]

Since $q_{1}=q^{1/4}$ we can easily see:

[TABLE]

where $u(\xi,z)$ and $P(\tau,z)$ are the ones in (2.6) and (2.4), respectively.

By (4.3) we see that $u_{1}(\xi,z)$ is convergent in a neighborhood of $(\xi,z)=(0,0)$. The equality (4.4) implies that $\lambda\in\mathbb{C}\setminus(\{0\}\cup S_{1})$ is equivalent to the condition $\lambda^{2}\in\mathbb{C}\setminus(\{0\}\cup S)$.

Since $\textrm{ord}_{t}(A_{j,\alpha})\geq 2j-2m_{0}+2$ holds for any $(j,\alpha)$ with $m_{0}\leq j<m$ and $|\alpha|>0$, the $q_{1}$-difference equation (4.1) satisfies the condition (2.5) (with $j$, $m_{0}$, $m$ replaced by $2j$, $2m_{0}$, $2m$, respectively). Therefore, we can apply (2) of Theorem 2.2 and its proof to the equation (4.1).

In particular, by the proof of [Proposition 5.6 in [11]] we have

Proposition 4.1**.**

For any $\rho\in\mathbb{C}\setminus(\{0\}\cup S_{1})$ we can find $\theta_{1}>0$ and $R_{1}>0$ which satisfy the following conditions (1) and (2):

(1)* $u_{1}(\xi,z)$ has an analytic extension $u^{*}_{1}(\xi,z)$ to the domain $S_{\theta_{1}}(\rho)\times D_{R_{1}}$.*

(2)* There are $\mu>0$ and holomorphic functions $w_{n}(\xi,z)$ ($n\geq\mu$) on $S_{\theta_{1}}(\rho)\times D_{R_{1}}$ which satisfy*

[TABLE]

and

[TABLE]

for some $A>0$ and $H>0$.

Therefore, by applying Proposition 3.2 to (4.5) we have the estimate

[TABLE]

for some $M>0$ and $\alpha\in\mathbb{R}$.

Completion of the proof of

Proposition 2.4.

Take any $\lambda=re^{\sqrt{-1}\theta}\in\mathbb{C}\setminus(\{0\}\cup S)$. We set $\rho=\sqrt{r}e^{\sqrt{-1}\theta/2}$: then we have $\rho\in\mathbb{C}\setminus(\{0\}\cup S_{1})$. Therefore, by Proposition 4.1 we can get $\theta_{1}>0$, $R_{1}>0$, $M>0$ and $\alpha\in\mathbb{R}$ such that $u_{1}(\xi,z)$ has an analytic extension $u_{1}^{*}(\xi,z)$ to the domain $S_{\theta_{1}}(\rho)\times D_{R_{1}}$ satisfying the estimate (4.6) on $S_{\theta_{1}}(\rho)\times D_{R_{1}}$.

Since $u_{1}(\xi,z)=u(q^{-1/4}\xi^{2},z)$ holds, this shows that $u(\xi,z)$ has also an analytic continuation $u^{*}(\xi,x)$ to the domain $S_{\theta}(\lambda)\times D_{R_{1}}$ (with $\theta=2\theta_{1}$), and we have $u^{*}(\xi,z)=u_{1}^{*}(q^{1/8}\xi^{1/2},z)$ on $S_{\theta}(\lambda)\times D_{R_{1}}$. Therefore, by (4.6) we have the estimate

[TABLE]

(with $M=M_{1}q^{1/32+\alpha/8}$ and $\beta=1/4+\alpha/2$).

Thus, by setting $\xi=\lambda q^{m}$ we obtain

[TABLE]

This proves (2.7). ∎