In this work, we draw attention to a family of singular linear $q-$difference differential equations modelled as for vanishing initial data $u(0,z)\equiv 0$, where $d_D,k_1\ge 1$ from the leading term of (1) are integers, $Q\left(X\right)$, $R_D\left(X\right)$ represent polynomials with complex coefficients and $P(t,z,V_1,V_2)$ stands for a polynomial in its arguments $t,V_1,V_2$ with holomorphic coefficients relatively to the space variable z on a horizontal strip in $\mathbb{C}$ designed as $H_{\beta }=\{z\in \mathbb{C}/|\mathrm{Im}\left(z\right)|<\beta \}$, for some prescribed width $2\beta >0$. The forcing term $f(t,z)$ is a logarithmic type function represented as a polynomial in both complex time variable t and inverse of its complex logarithm $1/logt$ with coefficients that are bounded holomorphic on the strip $H_{\beta }$.

This paper is a natural sequel of the recent study [9] by the author. Indeed, in [9], we focused on the next singularly perturbed linear partial differential equations shaped as for given initial data $y(0,z,ϵ)\equiv 0$, for integers $d_D,k_1\ge 1$ appearing in the principal term of (2), complex polynomials $Q\left(X\right)$, $R_D\left(X\right)$ as above and where $H(t,z,ϵ,V_1,V_2)$ represents a polynomial in $t,V_1,V_2$ whose coefficients are bounded holomorphic w.r.t z on the strip $H_{\beta }$ and relatively to a complex parameter $ϵ$ on some fixed disc $D_{{ϵ}_0}$ centered at 0 for some radius ${ϵ}_0>0$. The forcing term $h(t,z,ϵ)$ comprises coefficients that rely polynomially on complex time t, analytically in $ϵ$ on $D_{{ϵ}_0}$ and holomorphically in z on $H_{\beta }$. This term also entails logarithmic type functions stated as truncated Laplace transforms along a fixed segment $[-a,0]$ for some radius $a>0$ that involve the inverse complex logarithm $1/log\left(ϵt\right)$. When the radius $a>0$ is taken large, the expression of the forcing term h becomes proximate to maps that are similar to the forcing term $f(t,z)$ of (1) described above, namely a polynomial in both $ϵt$ and $1/log\left(ϵt\right)$ with bounded holomorphic coefficients on $D_{{ϵ}_0}\times H_{\beta }$.

Under suitable constraints set on the profile of (2), we were able to construct a set of genuine bounded holomorphic solutions $y_p$ to (2), for p in a finite subset $I_1$ of the natural numbers $\mathbb{N}$, expressed as a complete Laplace transform of integer order $k_1$ in the monomial $ϵt$, a truncated Laplace transform of order 1 in the inverse $1/log\left(ϵt\right)$ and inverse Fourier integral in the space variable z, where the so-called Borel-Fourier map $w_p({\tau }_1,{\tau }_2,m,ϵ)$ is

As a result, these functions $y_p(t,z,ϵ)$ define bounded holomorphic maps on domains $\mathcal{T}\times H_{\beta }\times {\mathcal{E}}_p$, for well selected bounded sector $\mathcal{T}$ edged at 0 and where $\underline{\mathcal{E}}={\left\{{\mathcal{E}}_p\right\}}_{p\in I_1}$ is an appropriate set of bounded sectors centered at 0. At this point, it is crucial to notice that these solutions $y_p$ cannot be represented as complete Laplace transform in the map $1/log\left(ϵt\right)$. It turns out that the radii $a,{ϵ}_0>0$ are related by a rule of the form ${ϵ}_0^{n_0}{a}^{n_1}\le M$, for some suitable constant $M>0$ and positive integers $n_0,n_1\ge 1$.

Besides, asymptotic features of these solutions have been examined in [9]. It appears that the family ${\left\{y_p\right\}}_{p\in I_1}$ owns asymptotic expansions of Gevrey type in two distinguished scales of functions. Indeed, for each $p\in I_1$, the partial map $(t,ϵ)\mapsto y_p(t,z,ϵ)$ holds a generalized asymptotic formal expansion (in a sense defined in the classical textbooks [5] and [14]) on the domain $\mathcal{T}\times {\mathcal{E}}_p$, in the scale of logarithmic functions ${\left\{{(1/log\left(ϵt\right))}^n\right\}}_{n\ge 0}$ with bounded holomorphic coefficients $G_{n,p}^1$ on $\mathcal{T}\times H_{\beta }\times {\mathcal{E}}_p$. These asymptotic expansions are revealed to be of Gevrey 1 on $\mathcal{T}\times {\mathcal{E}}_p$, giving rise to constants $K^1,M^1>0$ for which the error bounds occur for all integers $N\ge 0$, provided that $ϵ\in {\mathcal{E}}_p$, $t\in \mathcal{T}$, where $\Gamma \left(x\right)$ stands for the Gamma function in x. On the other hand, all the partial maps $ϵ\mapsto y_p(t,z,ϵ)$, $p\in I_1$, share a common generalized asymptotic formal expansion on ${\mathcal{E}}_p$, in the scale of monomial ${\left\{{ϵ}^n\right\}}_{n\ge 0}$ with bounded holomorphic coefficients $G_n^2$ on $\mathcal{T}\times H_{\beta }\times {\mathcal{D}}_{{ϵ}_0}$, for some open domain ${\mathcal{D}}_{{ϵ}_0}$ containing all the sectors ${\mathcal{E}}_p$, $p\in I_1$. Moreover, these asymptotic expansions happen to be of Gevrey order $1/k_1$ on each sector, meaning that constants $K_p^2,M_p^2>0$ can be pinpointed for which the error estimates hold for all integers $N\ge 0$, whenever $ϵ\in {\mathcal{E}}_p$. At last, we proved in [9] that the coefficients $G_{n,p}^1$ and $G_n^2$ of both formal expansions ${\widehat{y}}_p^1$ and ${\widehat{y}}^2$ solve explicit differential recursion relations with respect to $n\ge 0$ that might be handy for effective computations.

In the present investigation of the problem (1), we plan to follow a similar roadmap as in [9]. Namely, we plan to build up genuine sectorial solutions to (1) and describe their asymptotic expansions as time t borders the origin, instead of a perturbation parameter $ϵ$ which does not appear in (1). We notice that our main problem (1) can be viewed as a $q-$analog of (2) where the Fuchsian operator $t{\partial }_t$ is substituted by the discret dilation operator ${\sigma }_{q;t}$. This terminology stems from the plain observation that the quotient $\left(f\right(qt)-f(t\left)\right)/(qt-t)$ neighbors the derivative $f^{\prime }\left(t\right)$ as q tends to 1. The problem (2) involves at first sight only powers of the basic differential operator of Fuchsian type $t{\partial }_t$. However, the conditions imposed on (2) allows to express it also by means of powers of the basic differential operator of so-called irregular type $t^{k_1+1}{\partial }_t$. The same fact is acknowledged for the problem (1) under study for which $q-$difference operators of the form $t^{l_0}{\sigma }_{q;t}^{l_1}$ where $l_0\ge k_1{l}_1$ appear, see (22). These operators are labeled of irregular type in the literature by analogy with the differential case. We quote the classical textbooks [2] and [3] as references for analytic aspects of differential equations wih irregular type and the book [15] for analytic and algebraic features of $q-$difference equations with irregular type. This suggests that in the building process of the solutions to (1), the classical Laplace transform of order $k_1$ ought to be supplanted by a $q-$Laplace transform of order $k_1$ similarly to our earlier work [11] where some related initial value $q-$difference differential problem was handled.

We now describe a little more precisely the main statements of this paper achieved in Theorem 1 and Theorem 2. Namely, under fitting restrictions on the shape of (1) listed in Subsection 2.2 and complemented in the statement of Theorem 1 in Subsection 4.3, we can establish the existence of a bounded holomorphic solution $u(t,z)$ to (1) on a domain $(({\mathcal{R}}_{d_1,{\Delta }_1}\cap D_{R_1})\setminus (-\infty ,0])\times H_{\beta }$, for some small radius $R_1>0$, where ${\mathcal{R}}_{d_1,{\Delta }_1}$ stands for an open sector centered at 0 with large opening that does not contain the halfline $L_{d_1+\pi }=[0,+\infty )e^{\sqrt{-1}(d_1+\pi )}$, see (19), for thoroughly chosen directions $d_1\in \mathbb{R}$. In addition, the map $u(t,z)$ is modelled through a triple integral which entails a Fourier inverse, a $q-$Laplace and a complete Laplace transforms where the Borel-Fourier map ${\omega }_{d_1,\pi }({\tau }_1,{\tau }_2,m)$ is

At this stage, we emphasize that the geometry of the Borel space in the variable $({\tau }_1,{\tau }_2)$ for the map ${\omega }_{d_1,\pi }$ differs significantly from the one of the Borel-Fourier map $w_p$ in (3). Indeed, the map ${\omega }_{d_1,\pi }({\tau }_1,{\tau }_2,m)$ is in general not analytic near ${\tau }_1=0$ while $w_p({\tau }_1,{\tau }_2,m,ϵ)$ possesses this property. As we will realize later on, this will be the root of the dissemblances observed between the asymptotic properties of the solutions $y_p$ of (2) and the solution u of (1). Besides, the partial map ${\tau }_2\mapsto w_p({\tau }_1,{\tau }_2,m,ϵ)$ is only holomorphic on some fixed disc $D_a$ but ${\tau }_2\mapsto {\omega }_{d_1,\pi }({\tau }_1,{\tau }_2,m)$ is analytic on a full halfstrip $H_{\pi }$ which allows the solution $u(t,z)$ to be expressed as a complete Laplace transform in $1/logt$ in direction $\pi$ while $y_p(t,z,ϵ)$ is represented as a truncated Laplace transform along the segment $[-a,0]$. A direct by-product of this observation is that the forcing term $f(t,z)$ of (1) can be presented as an exact polynomial in both time t and inverse complex logarithm $1/logt$ while the forcing term $h(t,z,ϵ)$ has to be only considered as proximate to such a polynomial in t and $1/logt$. Some interesting aftermath is reached when $f(t,z)$ is chosen a mere monomial in t and $1/logt$ since in that case $f(t,z)$ solves an explicit nonlinear ordinary differential equation with polynomial coefficients in some positive rational power $t^{\alpha }$, $\alpha \in {\mathbb{Q}}_{+}$, displayed in (34). As a result, $u(t,z)$ turns out to be an exact holomorphic solution to some specific nonlinear $q-$difference differential equation with bounded holomorphic coefficients with respect to z on $H_{\beta }$ and polynomial in $t^{\alpha }$, stated in (36). Contrastingly, the equation (2) becomes close to some nonlinear partial differential equation as $a\to +\infty$ but no information can be extracted about the existence of an exact genuine solution to the limit nonlinear problem.

It is worthwhile noting that in the recent years much attention has been drawn on on nonlinear $q-$difference equations and especially on those related to the so-called $q-$Painlevé equations. For a comprehensive overview on major studies for $q-$Painlevé equations and more generally for integrable discrete dynamical systems, we refer to the book [7]. In this trend of research we quote the novel paper [6] where the authors construct convergent generalized power series with complex exponents on sectors that are solutions to nonlinear algebraic $q-$difference equations. In the context of nonlinear $q-$difference differential equations we mention an important result by H. Yamazawa obtained in [17]. Indeed, he considers equations with the shape for $t\in \mathbb{C}$, $x\in {\mathbb{C}}^n$, for some integers $n,m\ge 1$, some real number $q>1$, where F is a well prepared analytic function in its arguments. Under non resonance conditions of the so-called characteristic exponent $\rho \left(x\right)$ associated to (9) at $x=0$, he has constructed convergent logarithmic type solutions of the form where the coefficients $u_i\left(x\right)$ and ${\phi }_{i,j,k}\left(x\right)$ are holomorphic on a common disc $D_R$ and where ${\rho }_q\left(x\right)=log(1+(q-1)\rho \left(x\right))/log\left(q\right)$ stands for a $q-$analog of the characteristic exponent $\rho \left(x\right)$.

In the second part of Theorem 1, we exhibit for the solution $u(t,z)$ of (1) a generalized asymptotic expansion of Gevrey type in a logarithmic scale for t in the vicinity of 0. The statement is similar to the one reached in [9] for the solutions $y_p(t,z,ϵ)$ of (2). Indeed, the partial map $t\mapsto u(t,z)$ is shown to possess a generalized formal series with bounded holomorphic coefficients $G_n(t,z)$ on the domain $({\mathcal{R}}_{d_1,{\Delta }_1}\cap D_{R_1})\times H_{\beta }$ as asymptotic expansion of Gevrey order 1 with respect to t on $({\mathcal{R}}_{d_1,{\Delta }_1}\cap D_{R_1})\setminus (-\infty ,0]$, leading to estimates of the form for some constants $K,M>0$, for all integers $N\ge 0$, whenever $t\in ({\mathcal{R}}_{d_1,{\Delta }_1}\cap D_{R_1})\setminus (-\infty ,0]$. Furthermore, in Section 4.4, Proposition 6, we provide explicit and simple $q-$difference and differential recursion relations displayed in (154) and (155) for the coefficients $G_n(t,z)$, $n\ge 0$, intended for practical use. The existence of such a formal expression (10) is shown in a comparable way as (4) for the partial maps $(t,ϵ)\mapsto y_p(t,z,ϵ)$ in the problem (2). Namely, it is based on sharp estimates of some exponential decay for the differences of neighboring analytic solutions ${\left\{U_{d_1,{\mathfrak{d}}_p}(u_1,u_2,z)\right\}}_{0\le p\le \varsigma -1}$, disclosed in (120), to some related $q-$difference differential equation which comprises an homography action, see (116) and (118) in Proposition 4. In the process, we use a classical result known as the Ramis-Sibuya theorem (see Theorem (R.S.) in the subsection 4.2) which ensures the existence of a common Gevrey asymptotic expansion for families of sectorial holomorphic functions.

In the second main result of this paper, stated in Theorem 2, a generalized asymptotic expansion of the solution $u(t,z)$ is established in the scale of monomials ${\left\{t^n\right\}}_{n\ge 0}$. This statement differs notably from the one obtained for the partial maps $ϵ\mapsto y_p(t,z,ϵ)$ in the problem (2). Namely, the holomorphic solution $u(t,z)$ to (1) can be split into a sum $u(t,z)=u_1(t,z)+u_2(t,z)$ where

At this point, we stress the fact that the generalized expansion of Gevrey type (7) obtained for the solutions $y_p(t,z,ϵ)$ of (2) in the monomial scale ${\left\{{ϵ}^n\right\}}_{n\ge 0}$ are obtained by means of the Ramis-Sibuya theorem (see Theorem (R.S.) in Subsection 4.2) through precise estimates of some exponential decay for the differences of the consecutive maps $y_{p+1}-y_p$ relatively to $ϵ$ on the intersections ${\mathcal{E}}_{p+1}\cap {\mathcal{E}}_p$. These estimates were achieved according to the fact that the Borel-Fourier maps ${\tau }_1\mapsto w_p({\tau }_1,{\tau }_2,m,ϵ)$ are analytic at ${\tau }_1=0$ in (3). In contrast, for the problem (1) under study, as observed earlier in this introduction, any of the partial Borel-Fourier map ${\tau }_1\mapsto {\omega }_{d_1,\pi }({\tau }_1,{\tau }_2,m)$ appearing in (8) for any admissible direction $d_1\in \mathbb{R}$ is not analytic near ${\tau }_1=0$, only on sectors centered at 0. Therefore, no bounds for differences of solutions $u(t,z)$ to (1) for different directions $d_1\in \mathbb{R}$ can be rooted out and the recipe using the Ramis-Sibuya theorem fails to be applied. Instead we introduce a new approach based on a specific splitting of the triple integral (8) defining $u(t,z)$ and on the observation that the partial map ${\tau }_1\mapsto {\omega }_{d_1,\pi }({\tau }_1,{\tau }_2,m)$ can be analytically continued near ${\tau }_1=0$ provided that ${\tau }_2$ remains on the small disc $D_{\rho }$, see Proposition 10. Besides, whereas explicit differential recursions could be provided for the coefficients $G_n^2$, $n\ge 0$ of the formal expansions (6), no such relations are achieved for the coefficients $b_n(t,z)$, $n\ge 0$ of (12). However, explicit formulas (displayed in (207)) for $b_n$, $n\ge 0$, can be presented as double truncated $q-$Laplace, Laplace transforms and inverse Fourier integral of derivatives of the partial Borel-Fourier map ${\tau }_1\mapsto {\omega }_{d_1,\pi }({\tau }_1,{\tau }_2,m)$ at the origin.