On Inner Expansions for a Singularly Perturbed Cauchy Problem with Conﬂuent Fuchsian Singularities

: A nonlinear singularly perturbed Cauchy problem with conﬂuent Fuchsian singularities is examined. This problem involves coefﬁcients with polynomial dependence in time. A similar initial value problem with logarithmic reliance in time has recently been investigated by the author, for which sets of holomorphic inner and outer solutions were built up and expressed as a Laplace transform with logarithmic kernel. Here, a family of holomorphic inner solutions are constructed by means of exponential transseries expansions containing inﬁnitely many Laplace transforms with special kernel. Furthermore, asymptotic expansions of Gevrey type for these solutions relatively to the perturbation parameter are established.

D ,α (∂ t ) := ( t 2 − α )∂ t is a fuchsian differential operator at the points t = ± α− 1 2 for some odd integer α ≥ 3 and ∈ C * is a non-vanishing complex parameter. This operator unfolds the basic singularly perturbed irregular operator t 2 ∂ t of rank 1 at t = 0. Recent references about this so-called confluence process of Fuchsian singularities can be found in our work [1]. The function a( t, ) (unveiled in (13)) represents a well-chosen logarithmic function in its arguments. The linear differential operator P(T 1 , T 2 , T 3 , D ,α (∂ t ), ∂ z ) is chosen to be analytic in T 1 , T 3 near the origin in C and holomorphic w.r.t T 2 on a strip H β = {z ∈ C/|Im(z)| < β} for some width β > 0; moreover the forcing term f (T 1 , z, ) is analytic near the origin in C relatively to T 1 , and holomorphic in z on H β . Notice that this function a( t, ) is introduced to be able to construct nice representable solutions y(t, z, ) to (1) as Laplace transforms in time t from which parametric asymptotic properties can be analyzed. The fact that both coefficients and forcing term in (1) are holomorphic maps in 1/a( t, ) is a strong technical condition, but they turn out to be good approximations of general analytic functions on appropriate domains in time t, for z ∈ H β , provided that remains small enough.
Two distinguished finite sets of holomorphic solutions y(t, z, ) to (1) were constructed. The first family consists of the so-called outer solutions y out p (t, z, ), 0 ≤ p ≤ ι − 1 for some integer ι ≥ 2 that are holomorphic on domains A × H β × E out p , where A is a fixed bounded sectorial annulus confined apart of the origin in C and E out p is a bounded sector centered at 0 which belongs to a set E out = {E out p } 0≤p≤ι−1 that covers a full neighborhood of 0 in C * called good covering in C * (see Definition 2). The second family is comprised in the so-called inner solutions y in p (t, z, ), 0 ≤ p ≤ η − 1 for some integer η ≥ 2 that are constructed on a domain T in = { α−1 2 x/x ∈ χ} w.r.t time t for some fixed bounded sectorial annulus χ far enough from the origin in C, z ∈ H β and in a sector E in p that is part of a good covering in C * .
Both families y out/in p (t, z, ) could be expressed as special Laplace transform and Fourier integrals for suitable Borel maps W out/in p . Furthermore, from these integral representations, asymptotic behavior could be extracted. Indeed, the outer solutions y out p (t, z, ) (resp. inner solutions y in p (t, z, )) share a common power seriesÔ( ) = ∑ k≥0 O k k for bounded holomorphic coefficients O k on A × H β as asymptotic expansion of Gevrey order 1 (resp.Î( ) = ∑ k≥0 I k k for bounded holomorphic coefficients I k on T in × H β as asymptotic expansion of Gevrey order 2 α+1 ), meaning that constants A out/in p , B out/in p > 0 can be found with sup t∈A,z∈H β y out for all n ≥ 1, ∈ E out p and sup x∈χ,z∈H β y in p ( provided that n ≥ 1, ∈ E in p . In this paper, we turn our attention at a closely related singularly perturbed nonlinear Cauchy problem P 2 D ,α (∂ t ) ∂ S z u(t, z, ) = P 1 (t, z, , D ,α (∂ t ), ∂ z )u(t, z, ) + g(t, ) under given Cauchy data (∂ j z u)(t, 0, ) = ϕ j (t, ) , 0 ≤ j ≤ S − 1.
Like in (1), the forcing term g(t, ) is chosen to be a polynomial in the function 1/a( t, ). A similar assumption is put on the Cauchy data (6). The choice of the function a( t, ) is made for the same reason as in our former study (1). Moreover, P 2 stands for a polynomial with complex coefficients and S ≥ 2 is an integer. The novel feature is that the coefficients in the main part P 1 of (5) are assumed to be merely polynomials in t. As we will see in the work, this property has a deep impact on the structure of solutions to (5), (6) if compared with the ones y out/in p built up for (1). For technical reasons that will be clear later in the paper, we impose that the operator P 2 D ,α (∂ t ) can be factored out in the main part P 1 which allows us to reduce the problem (5), (6) to a Cauchy problem of Kowalevski type stated in (11), (12). Such a reduction is mandatory within our approach as explained in Section 5 of the work. The forcing term f (t, ) of the resulting Equation (11) is asked to solve a simple singularly perturbed ODE P 2 D ,α (∂ t ) f (t, ) = g(t, ). (7) As a result, the single Equation (5) is written as a coupling of a Kowalevski type Equation (11) and a singularly perturbed ODE (7). It is worthwhile noting that a more general forcing term g(t, z, ) relying holomorphically on z and Cauchy data ϕ j (t, ), 0 ≤ j ≤ S − 1 that are not only polynomials in 1/a( t, ) but also polynomials in t could be treated in a similar manner. However, such a general choice would lead to even more cumbersome and heavy computations which may avoid the reader to have a clear idea of the main purpose of the study.
The main striking difference with our previous work [1] is that now the analytic solutions fail to be constructed as a single special Laplace transform for the kernel exp τa( t, ) . Nevertheless, they can be expressed as exponential sums which involve infinitely many of Laplace transforms, which are called transseries in the literature (for an explanation of this terminology we refer to Chapters 4 and 5 of the excellent textbook [2]). Specifically, we can provide a finite set of analytic solutions (t, z) → u p (t, z, ), 0 ≤ p ≤ ι − 1 for some integer ι ≥ 2 on some domains T × D(0, r). Here, D(0, r) represents a small disc centered at 0 with radius r > 0 and T = { α−1 2 x/x ∈ χ 1 } (which is similar to the domain T in introduced above) is a set where χ 1 stands for a tiny bounded sectorial annulus close to 1 in C, whenever ∈ E p , where E p is a bounded sector centered at 0 that belongs to a good covering in C * . Since the domain T remains next to the Fuchsian singular point α−1 2 and borders the origin as tends to 0, we call the elements of this set inner solutions. Each solution u p (t, z, ) has a convergent exponential transseries expansion on T × D(0, r) for ∈ E p (see Theorem 1). Actually, the appearance of such transseries stems from the very expansion of the monomials ( t) l , l ≥ 1 as sums of special Laplace transforms as shown in Propositions 1 and 3. In the proof, an interesting small divisor phenomenon occurs which gives rise to the appearance of a special series (52) in the expansion of the basic monomial T, see (46). This special series turn out to carry an exponential series expansion displayed in Lemma 8. It is worthwhile noting that due to the specific arrangement of these expansions, this approach does not allow us to exhibit the so-called outer solutions as in the case of our previous study [1]. In a second main result (Theorem 2), we analyze the parametric asymptotic expansions of these inner solutions. It turns out that the functions → u p (t, z, ), 0 ≤ p ≤ ι − 1, share a common formal seriesÎ( ) = ∑ k≥0 I k k with bounded holomorphic coefficients I k on T × D(0, r) as Gevrey asymptotic expansion of order 2 α+1 . This outcome is comparable to the one obtained in [1].
During the last two decades, exponential transseries expansions appear to be a central tool in the study of differential, partial differential and difference equations in the complex domain. Indeed, we refer to the seminal work by O. Costin and R. Costin, see [3], where these class of expansions have shown to be essential in the study of formation of complex singularities along Stokes directions for systems of nonlinear ODEs of the form where y(x) ∈ C n , for an integer n ≥ 1, x ∈ C, A(x) is an analytic diagonal matrix and for a nonlinearity B(x, Y) analytic near the origin in C n+1 . Later, the transseries approach was extended by B. Braaksma and R. Kuik, in [4], to nonlinear systems of difference equations for y(x) ∈ C n , with integer n ≥ 1, Λ(x) an analytic diagonal matrix and g(x, Y) analytic near x = ∞ and Y = 0. Adjustments of this approach applied to partial differential equations (beyond the integrable case) have been initiated in the paper [5]. More recently, transseries expansions (conjointly with a KAM-like approach) have been applied to the location of complex singularities of general first order nonlinear scalar equations y = F(y(x), 1/x) as x tends to infinity, see [6]. Similar strategies have been implemented on nonlinear second order ODEs such as the Painlevé equation P 1 in order to compute in closed form connection coefficients between solutions on sectors called Stokes multipliers, see [7]. For problems related to obstruction for analytic integrability of Hamiltonian systems and transseries expansions of first integrals, we refer to [8]. Another aspect for which transseries turn out to be a powerful tool is the resurgence property of formal power series solutions to differential or more general functional equations (i.e., analytic continuation of their Borel transforms). For systems of the form (8) and (9) resurgence properties stemming from transseries expansions of actual holomorphic solutions on sectors have been exhibited in the papers [9,10]. For parametric resurgence for WKB solutions of 1D complex Schrödinger equations w.r.t , based on exponential series techniques, we mention the work by A. Fruchard and R. Schäfke [11]. Our paper is arranged as follows.
In Section 2, we present the main problems (11), (12) and (23), (12) of the work. In the technical Propositins 1 and 3, we express the coefficients of (11) as convergent exponential transseries expansions that contain Laplace transforms with specific kernel, which leads to seek for solutions to (11), (12) expressed in the same manner as exponential series involving Laplace transforms of infinitely many Borel maps W n , n ≥ 0. The principal accomplishment of this section is the layout of a sequence of convolution problems (100), (101) fulfilled by W n , n ≥ 0.
In Section 3, the sequence of convolution problems is solved by means of a majorant series approach, the use of sequences of Banach spaces of sectorial holomorphic functions with exponential growth and the application of the classical Cauchy-Kowalevski theorem.
In Section 4, the two main results of the paper are stated. A set of inner holomorphic solutions to (11), (23), (12) are constructed, which are defined w.r.t on a good covering in C * and relatively to time t on a domain that remains close to the moving fuchsian singularities of (23) (Theorem 1). In Theorem 2, the parametric asymptotic behavior of the latter solutions is analyzed by means of the classical Ramis-Sibuya approach.
The last section is devoted to the conclusion of the work where insights for prospective works are outlined.

Statement of the Main Problem and Related Auxiliary Equations
Let α ≥ 3 be an odd natural number. We set ∈ C * as a non-vanishing complex number. We define the differential operator D ,α (∂ t ) = ( t 2 − α )∂ t and for any integer l ≥ 1 we denote (D ,α (∂ t )) l the iteration of order l of the operator D ,α (∂ t ).
We consider a finite set I of N 3 , integers S, b ≥ 2 and ∆ l ≥ 0 that fulfill the next conditions for all l = (l 1 , l 2 , l 3 ) ∈ I. We state the main nonlinear Cauchy problem of our study ∂ S z u(t, z, ) = ∑ l=(l 1 ,l 2 ,l 3 )∈I for given Cauchy data The coefficients d l (z, ) and e(z, ) represent bounded holomorphic functions on a polydisc D(0, r) × D(0, 0 ) centered at the origin with radii r, 0 > 0. The forcing term f and the Cauchy data are displayed as follows. Let P 1 (τ) and P 2 (τ) two polynomials with complex coefficients. We assume that the set R 2 of roots of P 2 (τ) is located apart of some closed discD(0, ρ), for ρ > 0, and that P 1 (τ) is vanishing at 0. Let a(T, ) be the function already considered in our previous study [1], which represents a primitive of the rational function 1/(s 2 − α+1 ). We define the integral along a halfline L d = R + e √ −1d in direction d ∈ R which avoids the set R 2 and for which the integral is assumed to make sense. We set f (t, ) as a time rescaled version of F, namely The Cauchy data are built up in a similar manner. Specifically, for 0 ≤ j ≤ S − 1, let q j,n τ n be polynomials with complex coefficients that vanish at τ = 0. We introduce the integrals and set ϕ j (t, ) as a time rescaled version of Φ j , for 0 ≤ j ≤ S − 1. The domains where F, f , Φ j and ϕ j are well defined and holomorphic will be specified later in the work. In this work, the forcing term f is chosen in a way that it solves a simple ODE in the singular operator D ,α (∂ t ). Indeed, by construction of a(T, ), the action of D ,α (∂ t ) on f reads as On the other hand, a direct computation shows that for n ≥ 1, for any direction d for which the integral makes sense. If one expands P 1 (τ) = ∑ deg(P 1 ) n=1 p 1,n τ n , then we set g(t, ) = deg(P 1 ) ∑ n=1 p 1,n Γ(n)(−1) n (a( t, )) −n .
From (18) and (19), we deduce that f (t, ) solves the next singularly perturbed inhomogeneous ODE Moreover, the Cauchy data ϕ j (t, ) can be expressed as polynomials in the logarithmic function a( t, ) −1 , by using the representation (19), namely for any 0 ≤ j ≤ S − 1. As a result, let us introduce the next nonlinear differential map Then, if u(t, z, ) solves the main Equation (11), it also solves the next singularly perturbed nonlinear Cauchy problem under the Cauchy data (12), with forcing term g that represents a polynomial in the logarithmic function a( t, ) −1 . Throughout this work, we are searching for −time rescaled solutions of (11) with the shape u(t, z, ) = U( t, z, ).
As a matter of fact, if we define the next differential operator D ,α (∂ T ) = (T 2 − α+1 )∂ T the function U(T, z, ), by means of the change of variable T = t, is required to solve the next Cauchy problem assuming the Cauchy data We undertake a similar strategy as in our previous contribution [1], where solutions to the related problem (1) were asked to be expressed through special Laplace transforms that involve the function a(T, ) and Fourier integrals.
We first need to describe the domains in time T on which we plan to construct our solutions. Specifically, let us fix a bounded sectorial annulus (centered at 0) χ 0 = {x ∈ C/r 1 < |x| < r 2 , α 1 < arg(x) < α 2 } for some given radii r 1 , r 2 > 0 and with angles α 1 < α 2 . We set the next sectorial annulus (centered at 1) and introduce the next open sectorial domain for all ∈ C * . In the next lemma, bounds estimates are supplied for the map a(T, ).

Lemma 1.
For any given δ > 0, a small outer radius r 2 > 0 can be selected in a way that Since we can write is close to 0 provided that x ∈ χ 1 whenever 0 < r 1 < r 2 are taken small enough, Lemma 1 follows.
To be able to build up actual solutions of (24), our first main technical task will be here to express the most basic monomial T in terms of special Laplace transforms. The next Section 2.1 is devoted to the explanation of the next proposition.

Expression of a Monomial as Exponential Series with Special Laplace Transforms
We first need to introduce an analytic Borel transform for an analytic function near the origin outside a discrete set of pole singularities and analyze its Laplace transform accordingly. The next proposition turns out to be a variant of the proposition 12 of [12] which dealt with bounded sectorial analytic functions.

Proposition 2.
(1) Let ρ > 0 be a real number and let F : D(0, ρ) \ {0} → C be an analytic function apart of a discrete singular set Θ = {p n ∈ C * /n ∈ Z * } along which F possesses simple poles. We assume that

•
The set Θ is located on a line L 0 = l 0 R passing through 0 for some complex number l 0 ∈ C * and is contained in the open disc There exists a complex number F 0 such that for every open sector W + (resp. W − ) centered at 0, with radius We set the analytic Borel transform (of order 1) of F as the next integral along a positively oriented circle C(0, ρ/2) centered at 0 with radius ρ/2. The function (B 1 F)(z) defines an entire function on C and moreover we can find two constants C, M > 0 with for all z ∈ C.
(2) Let B : C → C be a holomorphic function with bounds for some given constants C, M > 0, for all z ∈ C. We define the Laplace transform (of order 1) of B in the direction d ∈ R as (3) For all n ∈ Z * , we denote Res u=p n (F(u)) the residue of the function F(u) at the pole u = p n . Assume that the next series converges on some subsector S ψ (centered at 0) of S(d, π,ρ). Then, the next identity holds for all T ∈ S ψ .
Proof. The proofs of the first two parts (1) and (2) are straightforward and can be performed by direct estimates on the integral representations. We focus on the third part (3). Using Fubini's theorem, we can write On the other hand, provided that T ∈ S ψ , the function u → F(u) u T T−u is holomorphic on D(0, ρ/2) \ {0} except at the singular points Θ ∪ {T}, for which the residue can be computed explicitly for all n ∈ Z * . By definition of the residue, and following a similar line of arguments as in the classical proof of the residue formula (see [13], Chap. 6), we can find a holomorphic function u → G T (u) on C except at Θ ∪ {T} (where it has simple poles) such that the function is holomorphic on the punctured disc D(0, ρ/2) \ {0} and such that By Cauchy's formula, one can deform the circle C(0, ρ/2) into any circle C(0, δ) centered at 0 with radius 0 < δ ≤ ρ/2 without changing the value of the integral, Now, we set ∆ 0 > 0 a small positive real number and we split the circle C(0, δ) into the union of four arcs of circles Accordingly, we decompose the integral along the circle into four parts From the hypotheses of the second item of 1), we can compute the next limit and in a similar manner we get On the other hand, since H T (u) is in particular continuous on the circle C(0, δ), we get that for j = 2, 4, all δ > 0 fixed. As a result, owing to the decomposition (40) and the above estimates (41), (42) together with (43), we observe that Therefore, according to (39), we reach that As a result, the identity (35) follows from (36), (38) and (44).
We now roughly explain the strategy which leads to the expansion (32). Assume that we can find a function b(u, ) that fulfills the next identity We denote Θ = {p n /n ∈ Z * } the set of poles of u → b(u, ). From the identity (35), we get the next decomposition If one replaces T by −1/a(T, ) in the latter decomposition, we can express T as a sum of a special Laplace transform of an entire function and a special series In a first step of the analysis, we compute b(u, ), describe its singular points and provide bounds w.r.t u and . We denote T → a −1 (T, ) the inverse function of T → a(T, ) that can be displayed as follows .
According to the requirement (45), we deduce that b(u, ) can be computed as We observe that the function u → b(u, ) is holomorphic in C * outside a discrete set of simple poles (which relies on ) given by and which is located on the line L 0, = √ −1 (α+1)/2 R passing through the origin.

Proof.
(1) We parametrize the circle C(0, ρ/2) by When u is taken on On the other hand, we remind that the convexity inequality | exp(z)| ≤ exp(|z|) holds for all z ∈ C and that there exists a holomorphic function ε(z) with e z = 1 + z + zε(z) such that ε(z) tends to 0 when z comes close to the origin in C. As a result, we get that provided that 0 > 0 is chosen small enough. The first part 1) of the lemma follows.
(2) We take u ∈ W + and v ∈ W − . By construction, we can write The second point (2) follows.
In the next lemma, we exhibit estimates for the analytic Borel transform of b(u, ).

Lemma 3.
We set as the analytic Borel transform of u → b(u, ). The function z → ω 1 (z, ) defines an entire function on C for all ∈ D(0, 0 ) \ {0}, moreover one can select two constants C, M > 0 (which rely on ρ and C 1 introduced in Lemma 2 (1) such that for all z ∈ C, all ∈ D(0, 0 ) \ {0}, provided that 0 is close enough to 0.
Proof. By definition of the analytic Borel transform (34) and the bounds (49), we get that In a second step, we focus on the special series We first need to compute the residue of the function u → b(u, ) at the singular points of the set Θ . Lemma 4. The residue of u → b(u, ) at the singular points p n of (48) are given by In the next lemma, we observe that each linear fractional map T → T p n (T−p n ) can be expressed as a Laplace transform through an explicit formula. However, the direction of integration depends on the sign of n.
for some ∆ + > 0 not too large (less than π/2). Then, the next formula (2) Let n ≤ −1 be a negative integer. We take a direction d − ∈ R with for some ∆ − > 0 not too large (less than π/2). Then, the next identity According to Lemmas 4 and 5, we notice that the special series ψ(T, ) can be expressed through the next limit formula for represent halflines at distance δ of the origin, whenever T is subjected to both (56) and (59).
In the last part of the proof, our goal will be to provide an explicit formula for the right-hand side of the identity (60) as exponential series involving Laplace transforms along the single direction d − . We set provided that |e τ/p 1 | < 1. The function τ → K 1 (τ, ) can be analytically extended on the whole plane C (as a function again denoted K 1 (τ, )) except at the discrete set of points We also define whenever τ ∈ L d − ,δ . The next lemma holds.
Proof. The first equality of (65) is a direct consequence of the uniform convergence of the series (61) (resp. (63)) on every compact segments of the halflines L d + ,δ (resp. L d − ,δ ).
We discuss now the second equality. We integrate the function τ → K 1 (τ, ) exp(−τ/T) along the oriented path formed by the union U d − d + ,δ of the segments −L d − ,δ , C d − d + ,δ and L d + ,δ . By construction, the path U d − d + ,δ encloses the set (which represents a subset of S described in (62)) of poles of the map τ → K 1 (τ, ) exp(−τ/T), under the constraint (64). The residue theorem implies that the next identity holds for all T ∈ C * under the restriction (56) and (59).
The next lemma computes the limit when δ tends to the origin of the integral along the arc of circle appearing in the right-hand side of the decomposition (65).
Proof. We parametrize the arc of circle which yields the equality From the expansion e z = 1 + z + zε(z) for a holomorphic function ε(z) near 0 such that lim z→0 ε(z) = 0, we get in particular that for any fixed ∈ D(0, 0 ) \ {0} and δ > 0 small enough. As a result, for any fixed ∈ D(0, 0 ) \ {0} we get that In the following closing lemma we provide an expression of the special series ψ(T, ) as an exponential series involving Laplace transforms.
Proof. The lemma is a direct consequence of (60) together with Lemmas 6 and 7.

Construction of a Family of Convolution Equations
Our next assignment is the expression of the fundamental building blocks T l in Equation (24) for any integer l ≥ 2 in terms of special Laplace transforms and exponential transseries. The next proposition holds.
At this stage of the proof, we can exhibit the shape of the solution to Equation (24) we are seeking for. Specifically, we assume that it can be expressed as an exponential transseries which involves infinitely many special Laplace transforms where L d − is the halfline described in Proposition 1. We assume that for each n ≥ 0, the expression (τ, z, ) → W n (τ, z, ) represents a function on a domain (S d − ∪ D(0, ρ)) × D(0, r) × D(0, 0 ) \ {0}, for an unbounded sector S d − with bisecting direction d − . We also assume that each integral along L d − composing U d − makes sense. From now on, we select the direction d in the formula (14) to be d = d − . We assume that the sector S d − and the direction d − are properly chosen in a way that S d − avoids the set R 2 of the roots of the polynomial P 2 (τ), for all ∈ D(0, 0 ) \ {0}.
In the remaining part of this subsection all the computations are made at a formal level. They are presented for the purpose to explain the reader how to derive a family of convolution equations that the sequence W n , n ≥ 0 is asked to solve in order that the expression U d − (T, z, ) fulfills the problem (24), (25). These computations will be justified and made rigorous later on (in the proof of Theorem 1), once we have shown that the resulting convolution Equations (100) and (101) possess actual holomorphic solutions subjected to the uniform bounds (149) (This will be the main objective of Section 3).
We first explain the action of the basic differential operator D ,α (∂ T ) on each term of the series (94). Indeed, according to the definition of a(T, ) given in (13), we get Furthermore, with the help of the transseries expansion of T l 1 displayed in (74), one can expand the next expression T l 1 (D ,α (∂ T )) l 2 ∂ l 3 z U d − (T, z, ) provided that l 1 ≥ 1. Specifically, Using Fubini's theorem we get where We now turn to the transseries expansions of the nonlinear term of (24). Indeed, Again, Fubini's theorem applies and allows us to write We are now ready to display the set of convolution equations which is asked to be fulfilled by the sequence (W n ) n≥0 .
The function W 0 (τ, z, ) is asked to fulfill the next nonlinear convolution equation For n ≥ 1, W n (τ, z, ) is required to solve the following linear convolution equation

Resolution of the Convolution Set of Equations within Banach Spaces of Holomorphic Functions
We seek for solutions W n (τ, z, ), n ≥ 0 of the convolution Equations (100) and (101) as formal power series w.r.t z, namely We first disclose a recursion formula, for each n ≥ 0, for the sequence of expressions W n,β (τ, ), β ≥ 0. We need to compute each piece of Equations (100) and (101). Specifically, for each n ≥ 0, we get Let the convergent Taylor expansion of d l (z, ) w.r.t z at 0 be for all ∈ D(0, 0 ). Owing to (104) and (105), we get and also On the other hand, we check that As a result, we require that the sequence of expressions W 0,β (τ, ), β ≥ 0 fulfills the next nonlinear recursive relation where δ 0,0 = 1 and δ 0,β = 0 whenever β ≥ 1, under the assumption that This latter constraint stems from the assumption (25) on the Cauchy data (∂ j z U d − )(T, 0, ) for 0 ≤ j ≤ S − 1. Furthermore, for each n ≥ 1, we ask the sequence of expressions W n,β (τ, ), for n ≥ 1, to be subjected to the next recursive relation 2 ) l 2 W q,β 2 +l 3 (τ, ) 2 ) l 2 W n,β 2 +l 3 (τ, ) that originates from our requirement (25) on the Cauchy data of our problem (24). We now need to specify in which spaces of functions our sequence of functions τ → W n,β (τ, ) are going to live, provided that ∈ D(0, 0 ) \ {0}. These Banach spaces have already been introduced in a former work of the author in [15]. Definition 1. Let S d be an unbounded sector centered at 0 with bisecting direction d ∈ R, D(0, ρ) be a disc centered at 0 with radius ρ > 0 and σ > 0 be a fixed real number. For each integer β ≥ 0, we set F (β,σ,S d ,ρ) as the vector space of holomorphic functions v : S d ∪ D(0, ρ) → C such that the norm ||v(τ)|| (β,σ,S d ,ρ) := sup is finite, where r b (β) represents the partial Riemann series r b (β) := ∑ β n≥0 1/(n + 1) b , for some integer b ≥ 2.
The next technical proposition turns to be essential in the discussion that will lead to the fact that the sequence of functions τ → W n,β (τ, ) actually belong to the space F (β,σ,S d − ,ρ) for the direction d − chosen as in Proposition 1 and for some small radius ρ > 0. Proposition 4. Take a real number σ such that σ > M for M given in (31). Select a radius ρ > 0 such that the disc D(0, ρ) does not contain any element of the set R 2 of the roots of the polynomial P 2 (τ).
Proof. We turn to the first point (1). By construction, both sets D(0, ρ) and S d − a properly chosen in a way that they avoid the roots R 2 of the polynomial P 2 (τ). We distinguish two cases.
At last, we discuss the third point (3). Owing to our assumption, we can control the functions W p,h 1 and W q,h 2 from above as follows and bearing in mind the definition of the constant F in (84), we deduce estimates for the convolution product provided that τ ∈ S d − ∪ D(0, ρ) and ∈ D(0, 0 ) \ {0}. This leads to (117).
In the sequel, we define the next sequence of numbers for all n, β ≥ 0.
According to the recursion (110) together with the constraints (111) and taking into account the estimates of Proposition 4, we obtain the next inequalities for the sequence W 0,β , β ≥ 0, for δ 0,0 = 1 and δ 0,β = 0 if β ≥ 1, under the condition that which are, by construction, finite positive numbers. Moreover, owing to the recursion (112) subjected to the conditions (113), with the help of the binomial expansion for 0 ≤ q ≤ n, the bounds of Proposition 4 allows us to get inequalities for the whole sequence W n,β , for any n ≥ 1, all β ≥ 0. Specifically, W n,β+S β! ≤ ∑ l=(l 1 ,l 2 ,l 3 )∈I,l 1 ≥1 under the additional condition that At this point of the proof, we plan to apply a majorant series method in order to be able to provide upper bounds for the whole sequence W n,β , for any integers n, β ≥ 0. Indeed, let us introduce a sequence of positive numbers W n,β for integers n, β ≥ 0 which are submitted to the next recursive relations.
For n = 0, the sequence W 0,β , β ≥ 0 fulfills for δ 0,0 = 1 and δ 0,β = 0 if β ≥ 1, strained to For any integer n ≥ 1, the sequence W n,β obeys the next rule for the given vanishing data We can check by induction the important fact that for any integers n, β ≥ 0. We build up the generating series Our next intention is to show that these series solve a Cauchy-Kowaleski type PDE, see (145), (146). Indeed, we define the next convergent series G 1,l 1 (X) = ∑ n≥0 K l 1 (L l 1 ) n X n , G 2,l 1 (X) = ∑ n≥0 A l 1 (B l 1 ) n X n and for l = (l 1 , l 2 , l 3 ) ∈ I. We now briefly describe the action of basic differential operators on W(X, Z), namely for integers l, m 1 , m 2 ≥ 1 and the product by convergent series From the recursions (134) Z W(X, Z)) + E(Z)F(W(X, Z)) 2 + P 1,2 (145) for given constant Cauchy data We now need to call upon the classical Cauchy-Kowalevski theorem (see [16], Chapter 1 for a reference), outlined below. Definition 2. Let ι ≥ 2 be an integer. We consider a finite set E = {E p } 0≤p≤ι−1 where E p stand for open sectors with vertex at 0 such that E p ⊂ D(0, 0 ) which fulfills the next three assumptions: The union of all the sectors E p covers a punctured disc centered at 0 in C.
Then, the set E is called a good covering in C * .
In the forthcoming definition, we describe the notion of admissible set of sectors relatively to a good covering.

Definition 3.
We consider a good covering E = {E p } 0≤p≤ι−1 in C * and a set of unbounded sectors S d p , 0 ≤ p ≤ ι − 1 with bisecting direction d p ∈ R that fulfill the next two properties: holds.
(2) There exists ∆ > 0 such that for all ∈ E p , one can choose a direction γ p ∈ R (that may depend on ) with In the first main statement of the work, we construct a family of actual holomorphic solutions to our main problems (11) and (23) under the Cauchy data (12). These solutions are defined on the sectors of a good covering E = {E p } 0≤p≤ι−1 in C * w.r.t the perturbation parameter . We control the difference between neighboring solutions on the intersection of sectors E p ∩ E p+1 where exponentially flat estimates are witnessed. Theorem 1. Assume that the requirement (10) on the shape of Equation (11) holds. We fix a good covering E = {E p } 0≤p≤ι−1 in C * together with an admissible set of sectors S = {S d p } 0≤p≤ι−1 relatively to E .
Then, for all ∈ E p , one can exhibit a solution (t, z) → u p (t, z, ) of the main problems (11) and (23) submitted to the Cauchy data (12) that remains bounded holomorphic on a domain T × D(0, r) provided that r, 0 > 0 are taken small enough. This solution is represented as an exponential transseries expansion which contains infinitely many special Laplace transforms where W n (τ, z, ) is the sequence of functions disclosed at the end of Section 3, for ∈ E p . Furthermore, the functions (x, z, ) → u p ( are bounded holomorphic on the domain χ 1 × D(0, r) × E p , for 0 ≤ p ≤ ι − 1. These functions suffer the next bounds: There exist constants K p , M p > 0 such that sup x∈χ 1 ,z∈D(0,r) for all ∈ E p ∩ E p+1 , for 0 ≤ p ≤ ι − 1 (where by convention u ι = u 0 ).

Proof.
In view of the feature (150) for the admissible set S, we remind the reader the construction we have reached at the end of Section 3. Specifically, we have singled out a sequence of functions W n (τ, z, ), n ≥ 0 satisfying the following property: Let 0 ≤ p ≤ ι − 1 and n ≥ 0 an integer, for all ∈ E p , the map (τ, z) → W n (τ, z, ) is holomorphic on the product (S d p ∪ D(0, ρ)) × D(0, r), provided that 0 < r ≤ 1/(2Z 1 ) and is subjected to the bounds (149) whenever τ ∈ S d p ∪ D(0, ρ) and z ∈ D(0, r).
For each 0 ≤ p ≤ ι − 1, we define the function where the integration halfline L γ p is chosen accordingly to Definition 3. We now show that when ∈ E p , the map (T, z) → U p (T, z, ) is well defined on the domain T × D(0, r) whenever the outer radius r 2 > 0 of χ 0 in (26) is taken small enough. Indeed, owing to the factorization (28), the next bounds hold According to the construction above, for each 0 ≤ p ≤ ι − 1, the function (t, z) → u p (t, z, ) is bounded, holomorphic and solves the main Cauchy problem (11), (12) (and hence the singularly perturbed Cauchy problem (23), (12)) on the domain T × D(0, r). Furthermore, the maps (x, z, ) → u p ( α−1 2 x, z, ) represent bounded holomorphic functions on the domain χ 1 × D(0, r) × E p , for all 0 ≤ p ≤ ι − 1.
Then, for all 0 ≤ p ≤ ι − 1, the functions G p ( ) share a common formal power seriesÎ( ) = ∑ k≥0 I k k where the coefficients I k belong to F, as Gevrey asymptotic expansion of order 1/k on E p . In other words, constants A p , B p > 0 can be selected with for all n ≥ 1, provided that ∈ E p .
We apply the above theorem to the set of functions for 0 ≤ p ≤ ι − 1, which represent holomorphic and bounded functions from E p into the Banach space F = O b (χ 1 × D(0, r)) equipped with the sup norm over χ 1 × D(0, r). Furthermore, the bounds (153) allow the cocycle Θ p ( ) = G p+1 ( ) − G p ( ) to fulfill the constraint (2) overhead. As a result, we deduce the existence of a formal power seriesÎ( ) that match the statement of Theorem 2.

Conclusions and Perspectives
In this work we have considered Cauchy problems which are conjointly singularly perturbed and possess confluent fuchsian singularities relying on a single perturbation parameter. We constructed genuine solutions expressed as exponential transseries expansions which are shown to appear naturally from the polynomial structure of the coefficients in the equations involved.
It turns out that the nature of these special solutions and the extraction of their parametric asymptotic behavior impose rather strong constraints on the shape of the equations under study in this work, namely the reduction to an equation of Kowalevski type displayed in (11). Concerning that point we may provide some explanatory comments that may help to investigate future lines of research in this topic. Indeed, departing from the singularly perturbed Equation (23) under less restrictive conditions on the differential operator P would lead to corresponding sets of convolution equations similar to (101) that ought to be solved relatively to τ (and on a good covering E ) on a domain where the expression P 2 (τ + n (α+1)/2 ) is not equal to zero. Here P 2 is the polynomial introduced in Section 2. However, for τ = 0, the quantity P 2 (n (α+1)/2 ) must vanish for a least some integer n ≥ 1 and some value of the parameter near 0. Consequently, the solution W n (τ, z, ) to the convolution equation may not be defined at τ = 0 and the solutions to (23) would not be constructible by means of Laplace transforms with special kernel. Hence, relaxing those constraints would need a new framework and novel ideas. It is worthwhile noting that strong restrictions on the shape of the coefficients are also asked in related works on confluence problems, see the references in our previous contribution [1].
At last, we expect that our approach can be adapted to other related problems, for instance in the context of q−difference or difference equations.
Funding: This research was partially funded by the University of Lille.

Conflicts of Interest:
The author declares no conflict of interest.