On Complex Singularity Analysis for Some Linear Partial Differential Equations in ℂ³

Abstract

We investigate the existence of local holomorphic solutions Y of linear partial differential equations in three complex variables whose coefficients are holomorphic on some polydisc in ℂ² outside some singular set Θ. The coefficients are written as linear combinations of powers of a solution X of some first-order nonlinear partial differential equation following an idea, we have initiated in a previous work (Malek and Stenger 2011). The solutions Y are shown to develop singularities along Θ with estimates of exponential type depending on the growth's rate of X near the singular set. We construct these solutions with the help of series of functions with infinitely many variables which involve derivatives of all orders of X in one variable. Convergence and bounds estimates of these series are studied using a majorant series method which leads to an auxiliary functional equation that contains differential operators in infinitely many variables. Using a fixed point argument, we show that these functional equations actually have solutions in some Banach spaces of formal power series.

1. Introduction

In this paper, we study a family of linear partial differential equations of the form 𝜕^𝑆_𝑤𝑌(𝑡,𝑧,𝑤)=_𝑘∈𝒮𝑎_1,𝑘(𝑡,𝑧,𝑤)𝜕_𝑡𝜕^𝑘_𝑤𝑌(𝑡,𝑧,𝑤)+𝑎_2,𝑘(𝑡,𝑧,𝑤)𝜕_𝑧𝜕^𝑘_𝑤𝑌(𝑡,𝑧,𝑤)+𝑎_3,𝑘(𝑡,𝑧,𝑤)𝜕^𝑘_𝑤𝑌(𝑡,𝑧,𝑤) (1) for given initial data 𝜕^𝑗_𝑤𝑌(𝑡,𝑧,0)=𝜑_𝑗(𝑡,𝑧), 0≤𝑗≤𝑆−1, where 𝒮 is a subset of ℕ² and 𝑆 is an integer which satisfies the constraints (175). The coefficients 𝑎_𝑚,𝑘(𝑡,𝑧,𝑤) are holomorphic functions on some domain (𝐷(0,𝑟)²⧵Θ)×𝐷(0,𝑤) where Θ is some singular set of 𝐷(0,𝑟)² (where 𝐷(0,𝛿) denotes the disc centered at 0 in ℂ with radius 𝛿>0) and the initial data 𝜑_𝑗(𝑡,𝑧) are assumed to be holomorphic functions on the polydisc 𝐷(0,𝑟)².

In order to avoid cumbersome statements and tedious computations, the authors have chosen to restrict their study to (1) that involves at most first-order derivatives with respect to 𝑡 and 𝑧 but the method proposed in this work can also be extended to higher order derivatives too.

In this work, we plan to construct holomorphic solutions of the problem (1) on (𝐷(0,𝑟)²⧵Θ)×𝐷(0,𝑤) and we will give precise growth rate for these solutions near the singular set Θ of the coefficients 𝑎_𝑚,𝑘(𝑡,𝑧,𝑤) (Theorem 21).

There exists a huge literature on the study of complex singularities and analytic continuation of solutions to linear partial differential equations starting from the fundamental contributions of Leray in [1]. Many important results are known for singular initial data and concern equations with bounded holomorphic coefficients. In that context, the singularities of the solution are generally contained in characteristic hypersurfaces issued from the singular locus of the initial conditions. For meromorphic initial data, we may refer to [2–5] and for more general ramified multivalued initial data, we may cite [6–9]. In our framework, the initial data are assumed to be nonsingular and the coefficients of the equation now carry the singularities. To the best knowledge of the authors, few results have been worked out in that case. For instance, the research of the so-called Fuchsian singularities in the context of partial differential equations is widely developed; we provide [10–13] as examples of references in this direction. It turns out that the situation we consider is actually close to a singular perturbation problem since the nature of the equation changes nearby the singular locus of it coefficients.

This work is a continuation of our previous study [14]. In [14], the authors focused on linear partial differential equations in ℂ². They have constructed local holomorphic solutions with a careful study of their asymptotic behaviour near the singular locus of the initial data. These initial data were chosen to be polynomial in 𝑡, 𝑧 and a function 𝑢(𝑡) satisfying some nonlinear differential equation of first order on some punctured disc 𝐷(𝑡₀,𝑟)⧵{𝑡₀}⊂ℂ and owning an isolated singularity at 𝑡₀ which is either a pole or an algebraic branch point according to a result of Painlevé. Inspired by the classical  tanh method introduced in [15], they have considered formal series solutions of the form 𝑢(𝑡,𝑧)=_𝑙≥0𝑢_𝑙(𝑡,𝑧)(𝑢(𝑡))^𝑙, (2) where 𝑢_𝑙 are holomorphic functions on 𝐷(𝑡₀,𝑟)×𝐷 where 𝐷⊂ℂ is a small disc centered at 0. They have given suitable conditions for these series to be well defined and holomorphic for 𝑡 in a sector 𝑆 with vertex 𝑡₀ and moreover as 𝑡 tends to 𝑡₀ the solutions 𝑢(𝑡,𝑧) are shown to carry at most exponential bounds estimates of the form 𝐶exp(𝑀|𝑡−𝑡₀|^−𝜇) for some constants 𝐶,𝑀,𝜇>0.

In this work, the coefficients 𝑎_𝑚,𝑘(𝑡,𝑧,𝑤) are constructed as polynomials in some function 𝑋(𝑡,𝑧) with holomorphic coefficients in (𝑡,𝑧,𝑤), where 𝑋(𝑡,𝑧) is now assumed to solve some nonlinear partial differential equation of first order and is asked to be holomorphic on a domain 𝐷(0,𝑟)²⧵Θ and to be singular along the set Θ. The class of functions in which one can choose the coefficients 𝑎_𝑚,𝑘(𝑡,𝑧,𝑤) is quite large since it contains meromorphic and multivalued holomorphic functions in (𝑡,𝑧) (see the example of Section 2.1).

In our setting, one cannot achieve the goal only dealing with formal expansions involving the function 𝑋(𝑡,𝑧) like (2) since the derivatives of 𝑋(𝑡,𝑧) with respect to 𝑡 or 𝑧 cannot be expressed only in terms of 𝑋(𝑡,𝑧). In order to get suitable recursion formulas, it turns out that we need to deal with series expansions that take into account all the derivatives of 𝑋(𝑡,𝑧) with respect to 𝑧. For this reason, the construction of the solutions will follow the one introduced in a recent work of Tahara and will involve Banach spaces of holomorphic functions with infinitely many variables.

In [16], Tahara introduced a new equivalence problem connecting two given nonlinear partial differential equations of first order in the complex domain. He showed that the equivalence maps have to satisfy the so-called coupling equations which are nonlinear partial differential equations of first order but with infinitely many variables. It is worthwhile saying that within the framework of mathematical physics, spaces of functions of infinitely many variables play a fundamental role in the study of nonlinear integrable partial differential equations known as solitons equations as described in the theory of Sato. See [17] for an introduction.

The layout of the paper is a follows. In a first step described in Section 2.2, we construct formal series of the form 𝑈(𝑡,𝑧,𝑤)=_𝛼≥0𝜙_𝛼𝜕𝑡,𝑧,^ℎ𝑧𝑋(𝑡,𝑧)ℎ!𝜈^ℎ_{0≤ℎ≤𝛼}𝑤^𝛼,𝛼! (3) solutions of some auxiliary nonhomogeneous integrodifferential equation (17) with polynomial coefficients in 𝑋(𝑡,𝑧). The coefficients 𝜙_𝛼, 𝛼≥0, are holomorphic functions on some polydisc in ℂ^𝛼+3 that satisfy some differential recursion (Proposition 2).

In Section 2.3, we establish a sequence of inequalities for the modulus of the differentials of arbitrary order of the functions 𝜙_𝛼 denoted by 𝜑_{𝛼,𝑛0,𝑛1,(𝑙ℎ)0≤ℎ≤𝛼} for all nonnegative integers 𝛼,𝑛₀,𝑛₁,𝑙_ℎ with 0≤ℎ≤𝛼 (Proposition 3). In the next section, we construct a sequence of coefficients 𝜓_{𝛼,𝑛0,𝑛1,(𝑙ℎ)0≤ℎ≤𝛼} which is larger than the latter sequence 𝜑_{𝛼,𝑛0,𝑛1,(𝑙ℎ)0≤ℎ≤𝛼}≤𝜓_{𝛼,𝑛0,𝑛1,(𝑙ℎ)0≤ℎ≤𝛼} (4) for any nonnegative integers 𝛼,𝑛₀,𝑛₁,𝑙_ℎ with 0≤ℎ≤𝛼 and whose generating formal series satisfies some integrodifferential functional equation (51) that involves differential operators with infinitely many variables (Propositions 5 and 6). The idea of considering recursions over the complete family of derivatives and the use of majorant series which lead to auxiliary Cauchy problems were already applied in former papers by the authors of this work; see [14, 18–21].

In Section 3, we solve the functional equation (51) by applying a fixed point argument in some Banach space of formal series with infinitely many variables (Proposition 19). The definition of these Banach spaces (Definition 7) is inspired from formal series spaces introduced in our previous work [14]. The core of the proof is based on continuity properties of linear integrodifferential operators in infinitely many variables explained in Section 3.1 and constitutes the most technical part of the paper.

Finally, in Section 4, we prove the main result of our work. Namely, we construct analytic functions 𝑌(𝑡,𝑧,𝑤), solutions of (1) for the prescribed initial data, defined on sets 𝐾×𝐷(0,𝑤) for any compact set 𝐾⊂𝐷(0,𝑟)²⧵Θ with precise bounds of exponential type in terms of the maximum value of |𝑋(𝑡,𝑧)| over 𝐾 (Theorem 21). The proof puts together all the constructions performed in the previous sections. More precisely, for some specific choice of the nonhomogeneous term in (17), a formal solution (3) of (17) gives rise to a formal solution 𝑌(𝑡,𝑧,𝑤) of (1) with the given initial data that can be written as the sum of the integral 𝜕_𝑤^−𝑆𝑈(𝑡,𝑧,𝑤) and a polynomial in 𝑤 having the initial data 𝜑_𝑗 as coefficients. Owing to the fact that the generating series of the sequence 𝜓_{𝛼,𝑛0,𝑛1,(𝑙ℎ)0≤ℎ≤𝛼}, solution of (51), belongs to the Banach spaces mentioned above, we get estimates for the holomorphic functions 𝜙_𝛼 with precise bounds of exponential type in terms of the radii of the polydiscs where they are defined; see (196). As a result, the formal solution 𝑈(𝑡,𝑧,𝑤) is actually convergent for 𝑤 near the origin and for (𝑡,𝑧) belonging to any compact set of 𝐷(0,𝑟)⧵Θ. Moreover, exponential bounds are achieved; see (197). The same properties then hold for 𝑌(𝑡,𝑧,𝑤).

2. Formal Series Solutions of Linear Integrodifferential Equations

2.1. Some Nonlinear Partial Differential Equation

We consider the following nonlinear partial differential equation: 𝜕_𝑡𝑋(𝑡,𝑧)=𝑎(𝑡,𝑧)𝜕_𝑧𝑋(𝑡,𝑧)+^𝑑_𝑝=0𝑎_𝑝(𝑡,𝑧)𝑋^𝑝(𝑡,𝑧), (5) where 𝑑≥2 is some integer and the coefficients 𝑎(𝑡,𝑧), 𝑎_𝑝(𝑡,𝑧) are holomorphic functions on some polydisc 𝐷(0,𝑅^′)²⊂ℂ² such that 𝑎_𝑑(𝑡,𝑧) is not identically equal to zero on 𝐷(0,𝑅^′)².

Notice that (5) can be solved by using the classical method of characteristics which is described in some classical textbooks like [22, page 118] or [23, page 100]. However, the solutions of (5) cannot in general be expressed in closed form. Nevertheless, we can mention some general results concerning qualitative properties of holomorphic solutions to (5) and even to more general first-order partial differential equations of the form 𝜕_𝑡𝑢(𝑡,𝑥)=𝐹𝑡,𝑥,𝑢(𝑡,𝑥),𝜕_𝑥𝑢(𝑡,𝑥) (6) for (𝑡,𝑥)∈ℂ×ℂ^𝑛 where 𝐹 is some holomorphic function and 𝑛≥1 an integer. For the construction of holomorphic functions to (6) with singularities located on some specific hypersurfaces (like {𝑡=0}), see [24, 25]. For the existence of local multivalued holomorphic solutions ramified around some singular sets, we may refer to [26, 27]. Concerning the study of the analytic continuation of singular solutions bounded on some hypersurface, we cite [28] and with prescribed upper estimates, we quote [29, 30].

In this work, we make the assumption that (5) has a holomorphic solution 𝑋(𝑡,𝑧) on 𝐷(0,𝑅^′)²⧵Θ where Θ is some set of 𝐷(0,𝑅^′)² (Θ will be called a singular set in the sequel).

In the next example, we show that a large class of functions can be obtained as solutions of equations of the form (5).

Example 1. Let 𝑛≥1 be an integer and let 𝑔∶𝐷(0,𝑅^′)²→ℂ be a holomorphic function which is not identically equal to zero. We consider 𝑎_𝑛+1()=1𝑡,𝑧𝑛𝜕_𝑧𝑔(𝑡,𝑧)−𝜕_𝑡𝑔(𝑡,𝑧) (7) which defines a holomorphic function on 𝐷(0,𝑅^′)². Then, the function 𝑋(𝑡,𝑧)=1/(𝑔(𝑡,𝑧))^1/𝑛 is a holomorphic solution of the equation 𝜕_𝑡𝑋(𝑡,𝑧)=𝜕_𝑧𝑋(𝑡,𝑧)+𝑎_𝑛+1(𝑡,𝑧)𝑋^𝑛+1(𝑡,𝑧) (8) on 𝐷(0,𝑅^′)²⧵Θ where Θ is the singular set defined by Θ={(𝑡,𝑧)∈𝐷(0,𝑅^′)²/𝑔(𝑡,𝑧)∈𝐿_𝜃} and 𝐿_𝜃 is some half-line ℝ₊𝑒^𝑖𝜃 with 𝜃∈ℝ depending on the choice of the determination of the logarithm.

2.2. Composition Series

Let 𝑋 be as in the previous subsection. In the following, we choose a compact subset 𝐾₀ with nonempty interior of 𝐷(0,𝑅)²⧵Θ for some 𝑅<𝑅^′ and we consider a real number 𝜌>1 such that sup_{(𝑡,𝑧)∈𝐾0}|𝑋(≤𝜌𝑡,𝑧)|2. (9) Let 𝐾⊊𝐾₀ be a compact set with nonempty interior Int(𝐾). From the Cauchy formula, there exists a real number 𝜈>0 such that sup_{(𝑡,𝑧)∈Int(𝐾)}𝜕^ℎ_𝑧𝑋(𝑡,𝑧)ℎ!𝜈^ℎ≤𝜌2 (10) for all integers ℎ≥0. For all integers 𝛼≥0, we denote 𝐼(𝛼)={0,…,𝛼}. We consider a sequence of functions 𝜙_𝛼(𝑣₀,𝑣₁,(𝑢_ℎ)_{ℎ∈𝐼(𝛼)}) which are holomorphic and bounded on the polydisc 𝐷(0,𝑅)²Π_{ℎ∈𝐼(𝛼)}𝐷(0,𝜌), for all 𝛼≥0.

We define the formal series in the 𝑤 variable as 𝑈(𝑡,𝑧,𝑤)=_𝛼≥0𝜙_𝛼𝜕𝑡,𝑧,^ℎ𝑧𝑋(𝑡,𝑧)ℎ!𝜈^ℎ_{ℎ∈𝐼(𝛼)}𝑤^𝛼.𝛼! (11) For all 𝛼≥0, we consider a holomorphic and bounded function 𝜔_𝛼(𝑣₀,𝑣₁,(𝑢_ℎ)_{ℎ∈𝐼(𝛼)}) on the product 𝐷(0,𝑅^′)²Π_{ℎ∈𝐼(𝛼)}𝐷(0,𝜌). We define the formal series 𝜔(𝑡,𝑧,𝑤)=_𝛼≥0𝜔_𝛼𝜕𝑡,𝑧,^ℎ𝑧𝑋(𝑡,𝑧)ℎ!𝜈^ℎ_{ℎ∈𝐼(𝛼)}𝑤^𝛼.𝛼! (12) Let 𝒮 be a finite subset of ℕ and let 𝑆≥1 be an integer which satisfies the property that 𝑆>𝑘 (13) for all 𝑘∈𝒮. For all 𝑘∈𝒮, 𝑚=1,2,3, and all integers 𝛼≥0, we define a function 𝑏_{𝑚,𝑘,𝛼}(𝑡,𝑧,𝑢₀) which is holomorphic on 𝐷(0,𝑅^′)²×ℂ and satisfies estimates of the following form. There exist two constants 𝐷_𝑚,𝑘>0, 𝐷_𝑚,𝑘>0 and an integer 𝑑_𝑚,𝑘≥0 such that sup_{|𝑡|<𝑅′,|𝑧|<𝑅′,|𝑢0|≤𝜌}𝑏_{𝑚,𝑘,𝛼}𝑡,𝑧,𝑢₀≤𝐷_𝑚,𝑘𝜌^{𝑑𝑚,𝑘}𝐷^𝛼_𝑚,𝑘𝛼! (14) for all 𝛼≥0, with all 𝜌≥1. In particular, each function 𝑢₀↦𝑏_{𝑚,𝑘,𝛼}(𝑡,𝑧,𝑢₀) is a polynomial of degree at most 𝑑_𝑚,𝑘 for all (𝑡,𝑧)∈𝐷(0,𝑅^′)². Finally, for all 𝑘∈𝒮, 𝑚=1,2,3, we consider the series 𝑏_𝑚,𝑘𝑡,𝑧,𝑢₀,𝑤=_𝛼≥0𝑏_{𝑚,𝑘,𝛼}𝑡,𝑧,𝑢₀𝑤^𝛼𝛼! (15) which define holomorphic functions on 𝐷(0,𝑅^′)²×ℂ×𝐷(0,𝑤), for any 0<𝑤≤1/𝐷_𝑚,𝑘.

Proposition 2. Assume that the sequence of functions (𝜙_𝛼)_𝛼≥0 satisfies the following recursion: 𝜙_𝛼𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼)}𝛼!=_{𝑘∈𝒮𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝑏_1,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀𝛼₁!×𝜕_𝑣0𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}𝛼₂!+_{𝛼𝑗∈𝐼2+𝑘−𝑆𝑙1+𝑙2=𝑗}𝜕^𝑙1_𝑣1𝑎𝑣₀,𝑣₁𝑙₁!𝜈^𝑙1𝑙₂+1𝜈𝑢_𝑙2+1+^𝑑_{𝑝=0𝑗0+⋯+𝑗𝑝=𝑗}𝜕^𝑗0_𝑣1𝑎_𝑝𝑣₀,𝑣₁𝑗₀!𝜈^𝑗0×Π^𝑝_𝑙=1𝑢_𝑗𝑙×𝜕_𝑢𝑗𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}𝛼₂!+_{𝑘∈𝒮𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝑏_2,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀𝛼₁!×𝜕_𝑣1𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}𝛼₂!+_{𝑗∈𝐼(𝛼2+𝑘−𝑆)}𝑗+1𝜈𝑢_𝑗+1×𝜕_𝑢𝑗𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}𝛼₂!+_{𝑘∈𝒮𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝑏_3,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀𝛼₁!×𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}𝛼₂!+𝜔_𝛼𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼)}𝛼! (16) for all 𝛼≥0, all 𝑣₀,𝑣₁∈𝐷(0,𝑅), all 𝑢_ℎ∈𝐷(0,𝜌), for ℎ∈𝐼(𝛼). Then, the formal series 𝑈(𝑡,𝑧,𝑤) satisfies the following integrodifferential equation: 𝑈(𝑡,𝑧,𝑤)=_𝑘∈𝒮𝑏_1,𝑘(𝑡,𝑧,𝑋(𝑡,𝑧),𝑤)𝜕_𝑡𝜕_𝑤^−𝑆+𝑘𝑈(𝑡,𝑧,𝑤)+𝑏_2,𝑘(𝑡,𝑧,𝑋(𝑡,𝑧),𝑤)𝜕_𝑧𝜕_𝑤^−𝑆+𝑘𝑈(𝑡,𝑧,𝑤)+𝑏_3,𝑘(𝑡,𝑧,𝑋(𝑡,𝑧),𝑤)𝜕_𝑤^−𝑆+𝑘𝑈()+𝜔()𝑡,𝑧,𝑤𝑡,𝑧,𝑤 (17) for all (𝑡,𝑧)∈Int(𝐾), where 𝜕_𝑤^−𝑚 denotes the 𝑚-iterate of the usual integration operator ∫^𝑤₀[⋅]𝑑𝑠.

Proof . We have that 𝑏_3,𝑘(𝑡,𝑧,𝑋(𝑡,𝑧),𝑤)𝜕_𝑤^−𝑆+𝑘𝑈(𝑡,𝑧,𝑤)=_{𝛼≥0𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝑏𝛼!^3,𝑘,𝛼1(𝑡,𝑧,𝑋(𝑡,𝑧))𝛼₁!×𝜙_{𝛼2+𝑘−𝑆}𝑡,𝑧,𝜕^ℎ_𝑧𝑋(𝑡,𝑧)/ℎ!𝜈^ℎ_{ℎ∈𝐼(𝛼2+𝑘−𝑆)}𝛼₂!×𝑤^𝛼,𝛼! (18) and we also see that 𝑏_2,𝑘(𝑡,𝑧,𝑋(𝑡,𝑧),𝑤)𝜕_𝑧𝜕_𝑤^−𝑆+𝑘𝑈(𝑡,𝑧,𝑤)=_{𝛼≥0𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝑏𝛼!^2,𝑘,𝛼1(𝑡,𝑧,𝑋(𝑡,𝑧))𝛼₁!×𝜕_𝑧𝜙_{𝛼2+𝑘−𝑆}𝑡,𝑧,𝜕^ℎ_𝑧𝑋(𝑡,𝑧)/ℎ!𝜈^ℎ_{ℎ∈𝐼(𝛼2+𝑘−𝑆)}𝛼₂!×𝑤^𝛼𝛼! (19) with 𝜕_𝑧𝜙_{𝛼2+𝑘−𝑆}𝜕𝑡,𝑧,^ℎ𝑧𝑋(𝑡,𝑧)ℎ!𝜈^ℎ_{ℎ∈𝐼(𝛼2+𝑘−𝑆)}=𝜕_𝑣1𝜙_{𝛼2+𝑘−𝑆}𝜕𝑡,𝑧,^ℎ𝑧𝑋(𝑡,𝑧)ℎ!𝜈^ℎ_{ℎ∈𝐼(𝛼2+𝑘−𝑆)}+_{𝛼𝑗∈𝐼2+𝑘−𝑆}𝜈𝜕𝑗+1^𝑧𝑗+1𝑋(𝑡,𝑧)𝑗+1!𝜈^𝑗+1𝜕_𝑢𝑗𝜙_{𝛼2+𝑘−𝑆}×𝜕𝑡,𝑧,^ℎ𝑧𝑋(𝑡,𝑧)ℎ!𝜈^ℎ_{ℎ∈𝐼(𝛼2+𝑘−𝑆)}, (20) for all (𝑡,𝑧)∈Int(𝐾). We also get that 𝑏_1,𝑘(𝑡,𝑧,𝑋(𝑡,𝑧),𝑤)𝜕_𝑡𝜕_𝑤^−𝑆+𝑘𝑈(𝑡,𝑧,𝑤)=_{𝛼≥0𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝑏𝛼!^1,𝑘,𝛼1(𝑡,𝑧,𝑋(𝑡,𝑧))𝛼₁!×𝜕_𝑡𝜙_{𝛼2+𝑘−𝑆}𝑡,𝑧,𝜕^ℎ_𝑧𝑋(𝑡,𝑧)/ℎ!𝜈^ℎ_{ℎ∈𝐼(𝛼2+𝑘−𝑆)}𝛼₂!×𝑤^𝛼𝛼! (21) with 𝜕_𝑡𝜙_{𝛼2+𝑘−𝑆}𝜕𝑡,𝑧,^ℎ𝑧𝑋(𝑡,𝑧)ℎ!𝜈^ℎ_{ℎ∈𝐼(𝛼2+𝑘−𝑆)}=𝜕_𝑣0𝜙_{𝛼2+𝑘−𝑆}𝜕𝑡,𝑧,^ℎ𝑧𝑋(𝑡,𝑧)ℎ!𝜈^ℎ_{ℎ∈𝐼(𝛼2+𝑘−𝑆)}+_{𝛼𝑗∈𝐼2+𝑘−𝑆}𝜕_𝑡𝜕^𝑗_𝑧𝑋(𝑡,𝑧)𝑗!𝜈^𝑗𝜕_𝑢𝑗𝜙_{𝛼2+𝑘−𝑆}×𝜕𝑡,𝑧,^ℎ𝑧𝑋(𝑡,𝑧)ℎ!𝜈^ℎ_{ℎ∈𝐼(𝛼2+𝑘−𝑆)}, (22) for all (𝑡,𝑧)∈Int(𝐾). Now, from (5) and the classical Schwarz’s result on equality of mixed partial derivatives, we get that 𝜕_𝑡𝜕^𝑗_𝑧𝑋(𝑡,𝑧)𝑗!𝜈^𝑗=𝜕^𝑗_𝑧𝜕_𝑡𝑋(𝑡,𝑧)𝑗!𝜈^𝑗=1𝑗!𝜈^𝑗𝜕^𝑗_𝑧𝑎(𝑡,𝑧)𝜕_𝑧𝑋(𝑡,𝑧)+^𝑑_𝑝=0𝑎_𝑝(𝑡,𝑧)𝑋^𝑝(𝑡,𝑧), (23) and from the Leibniz formula, we can write 1𝑗!𝜈^𝑗𝜕^𝑗_𝑧𝑎(𝑡,𝑧)𝜕_𝑧𝑋(𝑡,𝑧)=_{𝑙1+𝑙2=𝑗}𝜕^𝑙1_𝑧𝑎(𝑡,𝑧)𝑙₁!𝜈^𝑙1𝑙₂𝜈𝜕+1^𝑙2𝑧+1𝑋(𝑡,𝑧)𝑙₂+1!𝜈^𝑙2+1,1𝑗!𝜈^𝑗𝜕^𝑗_𝑧𝑎_𝑝(𝑡,𝑧)𝑋^𝑝()=𝑡,𝑧_{𝑗0+⋯+𝑗𝑝=𝑗}𝜕^𝑗0_𝑧𝑎_𝑝(𝑡,𝑧)𝑗₀!𝜈^𝑗0Π^𝑝_𝑙=1𝜕^𝑗𝑙_𝑧𝑋(𝑡,𝑧)𝑗_𝑙!𝜈^𝑗𝑙, (24) for all (𝑡,𝑧)∈Int(𝐾). Finally, gathering all the equalities above and using the recursion (16), one gets the integrodifferential equation (17).

2.3. Recursion for the Derivatives of the Functions 𝜙_𝛼, 𝛼≥0

We consider a sequence of functions 𝜙_𝛼(𝑣₀,𝑣₁,(𝑢_ℎ)_{ℎ∈𝐼(𝛼)}), 𝛼≥0, which are holomorphic and bounded on some polydisc 𝐷(0,𝑅)²Π_{ℎ∈𝐼(𝛼)}𝐷(0,𝜌) for some real numbers 𝑅>0 and 𝜌>1 and which satisfy the equalities (16). We introduce the sequence 𝜑_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}=sup_{|𝑣0|<𝑅,|𝑣1|<𝑅,|𝑢ℎ|<𝜌,ℎ∈𝐼(𝛼)}𝜕^𝑛0_𝑣0𝜕^𝑛1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ_𝑢ℎ×𝜙_𝛼𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼)} (25) for all 𝑛₀,𝑛₁≥0, all 𝑙_ℎ≥0, ℎ∈𝐼(𝛼), for all 𝛼≥0. We define also the following sequences: 𝑏_{𝑚,𝑘,𝛼,𝑛0,𝑛1,𝑙0}=sup_{|𝑣0|<𝑅,|𝑣1|<𝑅,|𝑢0|<𝜌}𝜕^𝑛0_𝑣0𝜕^𝑛1_𝑣1𝜕^𝑙0_𝑢0𝑏_{𝑚,𝑘,𝛼}𝑣₀,𝑣₁,𝑢₀,𝜔_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}=sup_{|𝑣0|<𝑅,|𝑣1|<𝑅,|𝑢ℎ|<𝜌,ℎ∈𝐼(𝛼)}𝜕^𝑛0_𝑣0𝜕^𝑛1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ_𝑢ℎ×𝜔_𝛼𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼)} (26) for 𝑚=1,2,3 and 𝑘∈𝒮. We put 𝐴_𝑗𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼+1)}=_{𝑙1+𝑙2=𝑗}𝜕^𝑙1_𝑣1𝑎𝑣₀,𝑣₁𝑙₁!𝜈^𝑙1𝑙₂+1𝜈𝑢_𝑙2+1+^𝑑_{𝑝=0𝑗0+⋯+𝑗𝑝=𝑗}𝜕^𝑗0_𝑣1𝑎_𝑝𝑣₀,𝑣₁𝑗₀!𝜈^𝑗0Π^𝑝_𝑙=1𝑢_𝑗𝑙,𝐵_𝑗𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼+1)}=𝑗+1𝜈𝑢_𝑗+1 (27)(28) for all 𝑗∈𝐼(𝛼), 𝑣₀,𝑣₁∈𝐷(0,𝑅^′) and 𝑢_ℎ∈ℂ, ℎ∈𝐼(𝛼). We define the sequences 𝐴_{𝑗,𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼+1)}=sup_{|𝑣0|<𝑅,|𝑣1|<𝑅,|𝑢ℎ|<𝜌,ℎ∈𝐼(𝛼)}𝜕^𝑛0_𝑣0𝜕^𝑛1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ_𝑢ℎ×𝐴_𝑗𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼+1)},𝐵_{𝑗,𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼+1)}=sup_{|𝑣0|<𝑅,|𝑣1|<𝑅,|𝑢ℎ|<𝜌,ℎ∈𝐼(𝛼)}𝜕^𝑛0_𝑣0𝜕^𝑛1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ_𝑢ℎ×𝐵_𝑗𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼+1)} (29) for all 𝑗∈𝐼(𝛼), all 𝑛₀,𝑛₁≥0, all 𝑙_ℎ≥0, ℎ∈𝐼(𝛼+1), for all 𝛼≥0. We also recall the definition of the Kronecker symbol 𝛿_0,𝑙 which is equal to 0 if 𝑙≠0 and equal to 1 if 𝑙=0.

Proposition 3. The sequence 𝜑_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)} satisfies the following inequality: 𝜑_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}𝛼!≤∑_𝑘∈𝒮∑_{𝛼12+𝛼=𝛼𝛼2≥𝑆−𝑘}∑_{𝑛010,1+𝑛0,2=𝑛,𝑛1,1+𝑛1,2=𝑛𝑙ℎ,1+𝑙ℎ,2=𝑙ℎ,ℎ∈𝐼(𝛼)}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,1!𝑛_0,2!𝑛_1,1!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑙_ℎ,2!×𝑏_{1,𝑘,𝛼1,𝑛0,1,𝑛1,1,𝑙0,1}𝛼₁!Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ,1×𝜑_{𝛼2+𝑘−𝑆,𝑛0,2+1,𝑛1,2,(𝑙ℎ,2)2ℎ∈𝐼(𝛼+𝑘−𝑆)}𝛼₂!Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ,2+∑_{𝑗∈𝐼(𝛼2+𝑘−𝑆)}∑_{𝑛010,1+𝑛0,2+𝑛0,3=𝑛,𝑛1,1+𝑛1,2+𝑛1,3=𝑛𝑙ℎ,1+𝑙ℎ,2+𝑙ℎ,3=𝑙ℎ,ℎ∈𝐼(𝛼)}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,1!𝑛_0,2!𝑛_0,3!𝑛_1,1!𝑛_1,2!𝑛_1,3!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑙_ℎ,2!𝑙_ℎ,3!×𝑏_{1,𝑘,𝛼1,𝑛0,1,𝑛1,1,𝑙0,1}𝛼₁!Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ,1×𝐴_{𝑗,𝛼2+𝑘−𝑆+1,𝑛0,2,𝑛1,2,(𝑙ℎ,2)2ℎ∈𝐼(𝛼+𝑘−𝑆+1)}×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆+1)}𝛿_0,𝑙ℎ,2×𝜑_{𝛼2+𝑘−𝑆,𝑛0,3,𝑛1,3,(𝑙ℎ,3)2ℎ∈𝐼(𝛼+𝑘−𝑆),ℎ≠𝑗,𝑙𝑗,3+1}𝛼₂!×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ,3+∑_𝑘∈𝒮∑_{𝛼12+𝛼=𝛼𝛼2≥𝑆−𝑘}∑_{𝑛010,1+𝑛0,2=𝑛,𝑛1,1+𝑛1,2=𝑛𝑙ℎ,1+𝑙ℎ,2=𝑙ℎ,ℎ∈𝐼(𝛼)}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,1!𝑛_0,2!𝑛_1,1!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑙_ℎ,2!×𝑏_{2,𝑘,𝛼1,𝑛0,1,𝑛1,1,𝑙0,1}𝛼₁!Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ,1×𝜑_{𝛼2+𝑘−𝑆,𝑛0,2,𝑛1,2+1,(𝑙ℎ,2)2ℎ∈𝐼(𝛼+𝑘−𝑆)}𝛼₂!Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ,2+∑_{𝑗∈𝐼(𝛼2+𝑘−𝑆)}∑_{𝑛010,1+𝑛0,2+𝑛0,3=𝑛,𝑛1,1+𝑛1,2+𝑛1,3=𝑛𝑙ℎ,1+𝑙ℎ,2+𝑙ℎ,3=𝑙ℎ,ℎ∈𝐼(𝛼)}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,1!𝑛_0,2!𝑛_0,3!𝑛_1,1!𝑛_1,2!𝑛_1,3!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑙_ℎ,2!𝑙_ℎ,3!×𝑏_{2,𝑘,𝛼1,𝑛0,1,𝑛1,1,𝑙0,1}𝛼₁!Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ,1×𝐵_{𝑗,𝛼2+𝑘−𝑆+1,𝑛0,2,𝑛1,2,(𝑙ℎ,2)2ℎ∈𝐼(𝛼+𝑘−𝑆+1)}×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆+1)}𝛿_0,𝑙ℎ,2×𝜑_{𝛼2+𝑘−𝑆,𝑛0,3,𝑛1,3,(𝑙ℎ,3)2ℎ∈𝐼(𝛼+𝑘−𝑆),ℎ≠𝑗,𝑙𝑗,3+1}𝛼₂!×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ,3+∑_𝑘∈𝒮∑_{𝛼12+𝛼=𝛼𝛼2≥𝑆−𝑘}∑_{𝑛010,1+𝑛0,2=𝑛,𝑛1,1+𝑛1,2=𝑛𝑙ℎ,1+𝑙ℎ,2=𝑙ℎ,ℎ∈𝐼(𝛼)}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,1!𝑛_0,2!𝑛_1,1!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑙_ℎ,2!×𝑏_{3,𝑘,𝛼1,𝑛0,1,𝑛1,1,𝑙0,1}𝛼₁!Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ,1×𝜑_{𝛼2+𝑘−𝑆,𝑛0,2,𝑛1,2,(𝑙ℎ,2)2ℎ∈𝐼(𝛼+𝑘−𝑆)}𝛼₂!Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ,2+𝜔_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}𝛼! (30) for all 𝛼≥0, all 𝑛₀,𝑛₁,𝑙_ℎ≥0 for ℎ∈𝐼(𝛼).

Proof. In order to get the inequality (30), we apply the differential operator 𝜕^𝑛0_𝑣0𝜕^𝑛1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ_𝑢ℎ on the left and right hand side of the recursion (16) and we use the expansions that are computed below.
From the Leibniz formula, we deduce that 𝜕^𝑛0_𝑣0𝜕^𝑛1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ_𝑢ℎ×𝑏_3,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}=_{𝑛010,1+𝑛0,2=𝑛,𝑛1,1+𝑛1,2=𝑛𝑙ℎ,1+𝑙ℎ,2=𝑙ℎ(𝛼),ℎ∈𝐼}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,1!𝑛_0,2!𝑛_1,1!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑙_ℎ,2!×𝜕^𝑛0,1_𝑣0𝜕^𝑛1,1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,1_𝑢ℎ𝑏_3,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀×𝜕^𝑛0,2_𝑣0𝜕^𝑛1,2_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,2_𝑢ℎ×𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)},𝜕^𝑛0_𝑣0𝜕^𝑛1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ_𝑢ℎ×𝑏_1,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀𝜕_𝑣0𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}=_{𝑛010,1+𝑛0,2=𝑛,𝑛1,1+𝑛1,2=𝑛𝑙ℎ,1+𝑙ℎ,2=𝑙ℎ(𝛼),ℎ∈𝐼}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,1!𝑛_0,2!𝑛_1,1!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑙_ℎ,2!×𝜕^𝑛0,1_𝑣0𝜕^𝑛1,1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,1_𝑢ℎ𝑏_1,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀×𝜕^𝑛0,2_𝑣⁺¹₀𝜕^𝑛1,2_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,2_𝑢ℎ×𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}. (31) Moreover, we can write 𝜕^𝑛0,1_𝑣0𝜕^𝑛1,1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,1_𝑢ℎ𝑏_3,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀=𝜕^𝑛0,1_𝑣0𝜕^𝑛1,1_𝑣1𝜕^𝑙0,1_𝑢0𝑏_3,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀×Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ,1 (32) with 𝜕^𝑛0,2_𝑣0𝜕^𝑛1,2_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,2_𝑢ℎ×𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}=𝜕^𝑛0,2_𝑣0𝜕^𝑛1,2_𝑣1Π_{ℎ∈𝐼(𝛼2+𝑘−𝑆)}𝜕^𝑙ℎ,2_𝑢ℎ×𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ,2,𝜕^𝑛0,1_𝑣0𝜕^𝑛1,1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,1_𝑢ℎ𝑏_1,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀=𝜕^𝑛0,1_𝑣0𝜕^𝑛1,1_𝑣1𝜕^𝑙0,1_𝑢0𝑏_1,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀×Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ,1 (33)(34) with 𝜕^𝑛0,2_𝑣⁺¹₀𝜕^𝑛1,2_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,2_𝑢ℎ×𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}=𝜕^𝑛0,2_𝑣⁺¹₀𝜕^𝑛1,2_𝑣1Π_{ℎ∈𝐼(𝛼2+𝑘−𝑆)}×𝜕^𝑙ℎ,2_𝑢ℎ𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ,2. (35) By construction, we have 𝐴_𝑗𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆+1)}=_{𝑙1+𝑙2=𝑗}𝜕^𝑙1_𝑣1𝑎𝑣₀,𝑣₁𝑙₁!𝜈^𝑙1𝑙₂+1𝜈𝑢_𝑙2+1+^𝑑_{𝑝=0𝑗0+⋯+𝑗𝑝=𝑗}𝜕^𝑗0_𝑣1𝑎_𝑝𝑣₀,𝑣₁𝑗₀!𝜈^𝑗0Π^𝑝_𝑙=1𝑢_𝑗𝑙 (36) for all 𝑗∈𝐼(𝛼₂+𝑘−𝑆). Again, by the Leibniz formula, we get that 𝜕^𝑛0_𝑣0𝜕^𝑛1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ_𝑢ℎ×𝑏_1,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀𝐴_𝑗𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆+1)}×𝜕_𝑢𝑗𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}=_{𝑛010,1+𝑛0,2+𝑛0,3=𝑛,𝑛1,1+𝑛1,2+𝑛1,3=𝑛𝑙ℎ,1+𝑙ℎ,2+𝑙ℎ,3=𝑙ℎ,ℎ∈𝐼(𝛼)}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!×𝑛_0,1!𝑛_0,2!𝑛_0,3!𝑛_1,1!𝑛_1,2!𝑛_1,3!×Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑙_ℎ,2!𝑙_ℎ,3!⁻¹×𝜕^𝑛0,1_𝑣0𝜕^𝑛1,1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,1_𝑢ℎ×𝑏_1,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀×𝜕^𝑛0,2_𝑣0𝜕^𝑛1,2_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,2_𝑢ℎ×𝐴_𝑗𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆+1)}×𝜕^𝑛0,3_𝑣0𝜕^𝑛1,3_𝑣1Π_{ℎ∈𝐼(𝛼),ℎ≠𝑗}𝜕^𝑙ℎ,3_𝑢ℎ𝜕^𝑙𝑗,3_𝑢⁺¹_𝑗×𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}. (37) Inside the formula (37), we can rewrite the relations (34) and 𝜕^𝑛0,2_𝑣0𝜕^𝑛1,2_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,2_𝑢ℎ𝐴_𝑗𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆+1)}=𝜕^𝑛0,2_𝑣0𝜕^𝑛1,2_𝑣1Π_{ℎ∈𝐼(𝛼2+𝑘−𝑆+1)}𝜕^𝑙ℎ,2_𝑢ℎ×𝐴_𝑗𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆+1)}×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆+1)}𝛿_0,𝑙ℎ,2 (38) with 𝜕^𝑛0,3_𝑣0𝜕^𝑛1,3_𝑣1Π_{ℎ∈𝐼(𝛼),ℎ≠𝑗}𝜕^𝑙ℎ,3_𝑢ℎ𝜕^𝑙𝑗,3_𝑢⁺¹_𝑗×𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}=𝜕^𝑛0,3_𝑣0𝜕^𝑛1,3_𝑣1Π_{ℎ∈𝐼(𝛼2+𝑘−𝑆),ℎ≠𝑗}𝜕^𝑙ℎ,3_𝑢ℎ𝜕^𝑙𝑗,3_𝑢⁺¹_𝑗×𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ,3. (39) In the same way, one gets the following equalities: 𝜕^𝑛0_𝑣0𝜕^𝑛1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ_𝑢ℎ𝑏_2,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀𝜕_𝑣1×𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}=_{𝑛010,1+𝑛0,2=𝑛,𝑛1,1+𝑛1,2=𝑛𝑙ℎ,1+𝑙ℎ,2=𝑙ℎ(𝛼),ℎ∈𝐼}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,1!𝑛_0,2!𝑛_1,1!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑙_ℎ,2!×𝜕^𝑛0,1_𝑣0𝜕^𝑛1,1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,1_𝑢ℎ𝑏_2,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀×𝜕^𝑛0,2_𝑣0𝜕^𝑛1,2_𝑣⁺¹₁Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,2_𝑢ℎ×𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)} (40) with the factorizations 𝜕^𝑛0,1_𝑣0𝜕^𝑛1,1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,1_𝑢ℎ𝑏_2,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀=𝜕^𝑛0,1_𝑣0𝜕^𝑛1,1_𝑣1𝜕^𝑙0,1_𝑢0𝑏_2,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀×Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ,1,𝜕^𝑛0,2_𝑣0𝜕^𝑛1,2_𝑣⁺¹₁Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,2_𝑢ℎ×𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}=𝜕^𝑛0,2_𝑣0𝜕^𝑛1,2_𝑣⁺¹₁Π_{ℎ∈𝐼(𝛼2+𝑘−𝑆)}𝜕^𝑙ℎ,2_𝑢ℎ×𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ,2. (41)(42) We recall that 𝐵_𝑗𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆+1)}=𝑗+1𝜈𝑢_𝑗+1 (43) for all 𝑗∈𝐼(𝛼₂+𝑘−𝑆) and we deduce that 𝜕^𝑛0_𝑣0𝜕^𝑛1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ_𝑢ℎ𝑏_2,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀×𝐵_𝑗𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆+1)}×𝜕_𝑢𝑗𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}=_{𝑛010,1+𝑛0,2+𝑛0,3=𝑛,𝑛1,1+𝑛1,2+𝑛1,3=𝑛𝑙ℎ,1+𝑙ℎ,2+𝑙ℎ,3=𝑙ℎ,ℎ∈𝐼(𝛼)}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!×𝑛_0,1!𝑛_0,2!𝑛_0,3!𝑛_1,1!𝑛_1,2!𝑛_1,3!×Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑙_ℎ,2!𝑙_ℎ,3!⁻¹×𝜕^𝑛0,1_𝑣0𝜕^𝑛1,1_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,1_𝑢ℎ×𝑏_2,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀×𝜕^𝑛0,2_𝑣0𝜕^𝑛1,2_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,2_𝑢ℎ×𝐵_𝑗𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆+1)}×𝜕^𝑛0,3_𝑣0𝜕^𝑛1,3_𝑣1Π_{ℎ∈𝐼(𝛼),ℎ≠𝑗}𝜕^𝑙ℎ,3_𝑢ℎ𝜕^𝑙𝑗,3_𝑢⁺¹_𝑗×𝜙_{𝛼2+𝑘−𝑆}𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆)}. (44) Inside the formula (44), we can rewrite the relations (41) and 𝜕^𝑛0,2_𝑣0𝜕^𝑛1,2_𝑣1Π_{ℎ∈𝐼(𝛼)}𝜕^𝑙ℎ,2_𝑢ℎ𝐵_𝑗𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆+1)}=𝜕^𝑛0,2_𝑣0𝜕^𝑛1,2_𝑣1Π_{ℎ∈𝐼(𝛼2+𝑘−𝑆+1)}𝜕^𝑙ℎ,2_𝑢ℎ×𝐵_𝑗𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2+𝑘−𝑆+1)}×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆+1)}𝛿_0,𝑙ℎ,2 (45) with the factorization (39).

2.4. Majorant Series and a Functional Equation with Infinitely Many Variables

Definition 4. One denotes by 𝔾[[𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊]] the vector space of formal series in the variables 𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,and𝑊 of the form Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0=,𝑊_𝛼≥0Ψ_𝛼𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)}𝑊^𝛼,𝛼! (46) where Ψ_𝛼∈ℂ[[𝑉₀,𝑉₁,(𝑈_ℎ)_{ℎ∈𝐼(𝛼)}]] for all 𝛼≥0.

We keep the notations of the previous section and we introduce the following formal series: 𝐵_𝑚,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊=_{𝛼≥0𝑛0,𝑛1,𝑙0≥0}𝑏_{𝑚,𝑘,𝛼,𝑛0,𝑛1,𝑙0}𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!𝑈^𝑙0₀𝑙₀!𝑊^𝛼,Ω𝑉𝛼!₀,𝑉₁,𝑈_ℎℎ≥0,𝑊=_{𝛼≥0𝑛0,𝑛1,𝑙ℎ(𝛼)≥0,ℎ∈𝐼}𝜔_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑙_ℎ!×𝑊^𝛼𝛼! (47) for 𝑚=1,2,3, all 𝑘∈𝒮, and 𝐀_𝑗,𝛼𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)}=_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)}𝐴_{𝑗,𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑙_ℎ!,𝐁_𝑗,𝛼𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)}=_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)}𝐵_{𝑗,𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑙_ℎ! (48) for all 𝛼≥0, all 𝑗∈𝐼(𝛼). We also introduce the following linear operators acting on 𝔾[[𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊]]. Let 𝔻_𝐀Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊=_{𝛼≥0(𝛼)𝑗∈𝐼}𝐀_𝑗,𝛼+1𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼+1)}×𝜕_𝑈𝑗Ψ_𝛼𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)}𝑊^𝛼,𝔻𝛼!_𝐁Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊=_{𝛼≥0(𝛼)𝑗∈𝐼}𝐁_𝑗,𝛼+1𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼+1)}×𝜕_𝑈𝑗Ψ_𝛼𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)}𝑊^𝛼𝛼! (49) for all Ψ∈𝔾[[𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊]]. We stress the fact that although these operators act on 𝔾[[𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊]] their image does not have to belong to this space.

Proposition 5. A formal series Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊=_{𝛼≥0𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)}𝜓_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑙_ℎ!×𝑊^𝛼𝛼! (50) satisfies the following functional equation: Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊=_𝑘∈𝒮𝐵_1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝜕_𝑉0Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊+𝐵_1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘×𝔻_𝐀Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊+_𝑘∈𝒮𝐵_2,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝜕_𝑉1Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊+𝐵_2,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘×𝔻_𝐁Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊+_𝑘∈𝒮𝐵_3,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0+Ω𝑉,𝑊₀,𝑉₁,𝑈_ℎℎ≥0,𝑊 (51) if and only if its coefficients 𝜓_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)} satisfy the following recursion: 𝜓_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}𝛼!=_{𝑘∈𝒮𝛼12+𝛼=𝛼𝛼2≥𝑆−𝑘𝑛010,1+𝑛0,2=𝑛,𝑛1,1+𝑛1,2=𝑛𝑙ℎ,1+𝑙ℎ,2=𝑙ℎ,ℎ∈𝐼(𝛼)}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,1!𝑛_0,2!𝑛_1,1!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑙_ℎ,2!×𝑏_{1,𝑘,𝛼1,𝑛0,1,𝑛1,1,𝑙0,1}𝛼₁!Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ,1×𝜓_{𝛼2+𝑘−𝑆,𝑛0,2+1,𝑛1,2,(𝑙ℎ,2)2ℎ∈𝐼(𝛼+𝑘−𝑆)}𝛼₂!Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ,2+_{𝑗∈𝐼(𝛼2+𝑘−𝑆)𝑛010,1+𝑛0,2+𝑛0,3=𝑛,𝑛1,1+𝑛1,2+𝑛1,3=𝑛𝑙ℎ,1+𝑙ℎ,2+𝑙ℎ,3=𝑙ℎ,ℎ∈𝐼(𝛼)}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,1!𝑛_0,2!𝑛_0,3!𝑛_1,1!𝑛_1,2!𝑛_1,3!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑙_ℎ,2!𝑙_ℎ,3!×𝑏_{1,𝑘,𝛼1,𝑛0,1,𝑛1,1,𝑙0,1}𝛼₁!Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ,1×𝐴_{𝑗,𝛼2+𝑘−𝑆+1,𝑛0,2,𝑛1,2,(𝑙ℎ,2)2ℎ∈𝐼(𝛼+𝑘−𝑆+1)}×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆+1)}𝛿_0,𝑙ℎ,2×𝜓_{𝛼2+𝑘−𝑆,𝑛0,3,𝑛1,3,(𝑙ℎ,3)2ℎ∈𝐼(𝛼+𝑘−𝑆),ℎ≠𝑗,𝑙𝑗,3+1}𝛼₂!×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ,3+_{𝑘∈𝒮𝛼12+𝛼=𝛼𝛼2≥𝑆−𝑘𝑛010,1+𝑛0,2=𝑛,𝑛1,1+𝑛1,2=𝑛𝑙ℎ,1+𝑙ℎ,2=𝑙ℎ,ℎ∈𝐼(𝛼)}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,1!𝑛_0,2!𝑛_1,1!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑙_ℎ,2!×𝑏_{2,𝑘,𝛼1,𝑛0,1,𝑛1,1,𝑙0,1}𝛼₁!Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ,1×𝜓_{𝛼2+𝑘−𝑆,𝑛0,2,𝑛1,2+1,(𝑙ℎ,2)2ℎ∈𝐼(𝛼+𝑘−𝑆)}𝛼₂!Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ,2+_{𝑗∈𝐼(𝛼2+𝑘−𝑆)𝑛010,1+𝑛0,2+𝑛0,3=𝑛,𝑛1,1+𝑛1,2+𝑛1,3=𝑛𝑙ℎ,1+𝑙ℎ,2+𝑙ℎ,3=𝑙ℎ,ℎ∈𝐼(𝛼)}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,1!𝑛_0,2!𝑛_0,3!𝑛_1,1!𝑛_1,2!𝑛_1,3!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑙_ℎ,2!𝑙_ℎ,3!×𝑏_{2,𝑘,𝛼1,𝑛0,1,𝑛1,1,𝑙0,1}𝛼₁!Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ,1×𝐵_{𝑗,𝛼2+𝑘−𝑆+1,𝑛0,2,𝑛1,2,(𝑙ℎ,2)2ℎ∈𝐼(𝛼+𝑘−𝑆+1)}×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆+1)}𝛿_0,𝑙ℎ,2×𝜓_{𝛼2+𝑘−𝑆,𝑛0,3,𝑛1,3,(𝑙ℎ,3)2ℎ∈𝐼(𝛼+𝑘−𝑆),ℎ≠𝑗,𝑙𝑗,3+1}𝛼₂!×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ,3+_{𝑘∈𝒮𝛼12+𝛼=𝛼𝛼2≥𝑆−𝑘𝑛010,1+𝑛0,2=𝑛,𝑛1,1+𝑛1,2=𝑛𝑙ℎ,1+𝑙ℎ,2=𝑙ℎ,ℎ∈𝐼(𝛼)}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,1!𝑛_0,2!𝑛_1,1!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑙_ℎ,2!×𝑏_{3,𝑘,𝛼1,𝑛0,1,𝑛1,1,𝑙0,1}𝛼₁!Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ,1×𝜓_{𝛼2+𝑘−𝑆,𝑛0,2,𝑛1,2,(𝑙ℎ,2)2ℎ∈𝐼(𝛼+𝑘−𝑆)}𝛼₂!Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ,2+𝜔_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}𝛼! (52) for all 𝛼≥0, all 𝑛₀,𝑛₁,𝑙_ℎ≥0 with ℎ∈𝐼(𝛼).

Proof. We proceed by identification of the coefficients in the Taylor expansion with respect to the variables 𝑉₀,𝑉₁,(𝑈_ℎ)_{ℎ∈𝐼(𝛼)}, and 𝑊 for all 𝛼≥0. By definition, we have that 𝐵_1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝜕_𝑉0Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊=_{𝛼≥0𝛼12+𝛼=𝛼𝛼2≥𝑆−𝑘}𝒞¹_𝛼1,𝛼2𝑊^𝛼, (53) where the coefficients 𝒞¹_𝛼1,𝛼2 can be rewritten, using the Kronecker symbol 𝛿_0,𝑚, in the form 𝒞¹_𝛼1,𝛼2=_{𝑛0,𝑛1,𝑙ℎ(𝛼)≥0,ℎ∈𝐼}𝑏_{1,𝑘,𝛼1,𝑛0,𝑛1,𝑙0}𝛼₁!×Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑙_ℎ!×_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)}𝜓_{𝛼2+𝑘−𝑆,𝑛0+1,𝑛1,(𝑙ℎ)2ℎ∈𝐼(𝛼+𝑘−𝑆)}𝛼₂!×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑙_ℎ!. (54) Hence, 𝒞¹_𝛼1,𝛼2=_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)𝑛010,1+𝑛0,2=𝑛,𝑛1,1+𝑛1,2=𝑛𝑙ℎ,1+𝑙ℎ,2=𝑙ℎ,ℎ∈𝐼(𝛼)}𝑏_{1,𝑘,𝛼1,𝑛0,1,𝑛1,1,𝑙0,1}𝛼₁!𝑛_0,1!𝑛_1,1!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!×Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ,1×𝜓_{𝛼2+𝑘−𝑆,𝑛0,2+1,𝑛1,2,(𝑙ℎ,2)2ℎ∈𝐼(𝛼+𝑘−𝑆)}𝛼₂!𝑛_0,2!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,2!×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ,2×𝑉^𝑛0₀𝑉^𝑛1₁Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ. (55) We also have that 𝐵_1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝔻_𝐀Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊=_{𝛼≥0𝛼12+𝛼=𝛼𝛼2≥𝑆−𝑘}ℱ¹_𝛼1,𝛼2𝑊^𝛼, (56) where the coefficients ℱ¹_𝛼1,𝛼2 can be rewritten in the form ℱ¹_𝛼1,𝛼2=_{𝛼𝑗∈𝐼2−𝑆+𝑘𝑛0,𝑛1,𝑙ℎ(𝛼)≥0,ℎ∈𝐼}𝑏_{1,𝑘,𝛼1,𝑛0,𝑛1,𝑙0}𝛼₁!×Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ×𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑙_ℎ!×_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)}𝐴_{𝑗,𝛼2−𝑆+𝑘+1,𝑛0,𝑛1,(𝑙ℎ)2ℎ∈𝐼(𝛼−𝑆+𝑘+1)}×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2−𝑆+𝑘+1)}𝛿_0,𝑙ℎ×𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑙_ℎ!×_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)}𝜓_{𝛼2−𝑆+𝑘,𝑛0,𝑛1,(𝑙ℎ)2ℎ∈𝐼(𝛼−𝑆+𝑘),ℎ≠𝑗,𝑙𝑗+1}𝛼₂!×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2−𝑆+𝑘)}𝛿_0,𝑙ℎ×𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑙_ℎ!. (57) Therefore, ℱ¹_𝛼1,𝛼2=∑_{𝑗∈𝐼(𝛼2−𝑆+𝑘)}∑_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)}∑_{𝑛010,1+𝑛0,2+𝑛0,3=𝑛,𝑛1,1+𝑛1,2+𝑛1,3=𝑛𝑙ℎ,1+𝑙ℎ,2+𝑙ℎ,3=𝑙ℎ,ℎ∈𝐼(𝛼)}𝑏_{1,𝑘,𝛼1,𝑛0,1,𝑛1,1,𝑙0,1}𝛼₁!𝑛_0,1!𝑛_1,1!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ,1×𝐴_{𝑗,𝛼2−𝑆+𝑘+1,𝑛0,2,𝑛1,2,(𝑙ℎ,2)2ℎ∈𝐼(𝛼−𝑆+𝑘+1)}𝑛_0,2!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,2!Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2−𝑆+𝑘+1)}𝛿_0,𝑙ℎ,2×𝜓_{𝛼2−𝑆+𝑘,𝑛0,3,𝑛1,3,(𝑙ℎ,3)2ℎ∈𝐼(𝛼−𝑆+𝑘),ℎ≠𝑗,𝑙𝑗,3+1}𝛼₂!𝑛_0,3!𝑛_1,3!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,3!Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2−𝑆+𝑘)}𝛿_0,𝑙ℎ,3×𝑉^𝑛0₀𝑉^𝑛1₁Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ. (58)
On the other hand, using similar computations we get 𝐵_2,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝜕_𝑉1Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊=_{𝛼≥0𝛼12+𝛼=𝛼𝛼2≥𝑆−𝑘}𝒞²_𝛼1,𝛼2𝑊^𝛼, (59) where 𝒞²_𝛼1,𝛼2=_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)𝑛010,1+𝑛0,2=𝑛,𝑛1,1+𝑛1,2=𝑛𝑙ℎ,1+𝑙ℎ,2=𝑙ℎ,ℎ∈𝐼(𝛼)}𝑏_{2,𝑘,𝛼1,𝑛0,1,𝑛1,1,𝑙0,1}𝛼₁!𝑛_0,1!𝑛_1,1!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!×Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ,1×𝜓_{𝛼2+𝑘−𝑆,𝑛0,2,𝑛1,2+1,(𝑙ℎ,2)2ℎ∈𝐼(𝛼+𝑘−𝑆)}𝛼₂!𝑛_0,2!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,2!×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ,2×𝑉^𝑛0₀𝑉^𝑛1₁Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ. (60) We also have that 𝐵_2,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝔻_𝐁Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊=_{𝛼≥0𝛼12+𝛼=𝛼𝛼2≥𝑆−𝑘}ℱ²_𝛼1,𝛼2𝑊^𝛼, (61) where ^{ℱ2𝛼1,𝛼2=∑𝑗∈𝐼(𝛼2−𝑆+}𝐵^{𝑘)∑𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)∑𝑛010,1+𝑛0,2+𝑛0,3=𝑛,𝑛1,1+𝑛1,2+𝑛1,3=𝑛𝑙ℎ,1+𝑙ℎ,2+𝑙ℎ,3=𝑙ℎ,ℎ∈𝐼(𝛼)𝑏2,𝑘,𝛼1,𝑛0,1,𝑛1,1,𝑙0,1𝛼1!𝑛0,1!𝑛1,1!Πℎ∈𝐼(𝛼)𝑙ℎ,1!Πℎ∈𝐼(𝛼)⧵{0}𝛿0,𝑙ℎ,1×𝐵𝑗,𝛼2−𝑆+𝑘+1,𝑛0,2,𝑛1,2,(𝑙ℎ,2)2ℎ∈𝐼(𝛼−𝑆+𝑘+1)𝑛0,2!𝑛1,2!Πℎ∈𝐼(𝛼)𝑙ℎ,2!Πℎ∈𝐼(𝛼)⧵𝐼(𝛼2−𝑆+𝑘+1)𝛿0,𝑙ℎ,2×𝜓𝛼2−𝑆+𝑘,𝑛0,3,𝑛1,3,(𝑙ℎ,3)2ℎ∈𝐼(𝛼−𝑆+𝑘),ℎ≠𝑗,𝑙𝑗,3+1𝛼2!𝑛0,3!𝑛1,3!Πℎ∈𝐼(𝛼)𝑙ℎ,3!Πℎ∈𝐼(𝛼)⧵𝐼(𝛼2−𝑆+𝑘)𝛿0,𝑙ℎ,3×𝑉𝑛00𝑉𝑛11Πℎ∈𝐼(𝛼)𝑈𝑙ℎℎ,}_3,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊=_{𝛼≥0𝛼12+𝛼=𝛼𝛼2≥𝑆−𝑘}𝒞³_𝛼1,𝛼2𝑊^𝛼, (62)(63) where 𝒞³_𝛼1,𝛼2=_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)𝑛010,1+𝑛0,2=𝑛,𝑛1,1+𝑛1,2=𝑛𝑙ℎ,1+𝑙ℎ,2=𝑙ℎ,ℎ∈𝐼(𝛼)}𝑏_{3,𝑘,𝛼1,𝑛0,1,𝑛1,1,𝑙0,1}𝛼₁!𝑛_0,1!𝑛_1,1!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!×Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ,1×𝜓_{𝛼2+𝑘−𝑆,𝑛0,2,𝑛1,2,(𝑙ℎ,2)2ℎ∈𝐼(𝛼+𝑘−𝑆)}𝛼₂!𝑛_0,2!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,2!×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2+𝑘−𝑆)}𝛿_0,𝑙ℎ,2×𝑉^𝑛0₀𝑉^𝑛1₁Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ. (64) Finally, gathering the expansions (55), (58), (60), and (62) with (64) yields the result.

Proposition 6. The sequences 𝜑_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)} and 𝜓_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)} satisfy the following inequalities: 𝜑_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}≤𝜓_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)} (65) for all 𝛼≥0, all 𝑛₀,𝑛₁≥0, all 𝑙_ℎ≥0, ℎ∈𝐼(𝛼).

Proof. For 𝛼=0, using the recursions (16) and (52), we get that 𝜑_{0,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(0)}=𝑤_{0,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(0)}=𝜓_{0,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(0)} (66) for all 𝑛₀,𝑛₁,𝑙₀≥0. By induction on 𝛼 and using the inequalities (30) together with the equalities (52), one gets the result.

3. Convergent Series Solutions for a Functional Equation with Infinitely Many Variables

3.1. Banach Spaces of Formal Series

Let 𝜌>1 and 𝜎,𝑉₀,𝑉₁,𝑊,𝛿>0 be real numbers. For any given real number 𝑏>1, we define the sequences 𝑟_𝑏(𝛼)=∑^𝛼_𝑛=01/(𝑛+1)^𝑏 for all 𝛼≥0 and 𝑈_ℎ=𝛿/(ℎ^𝑏+1) for all ℎ≥0.

Definition 7. Let 𝛼≥0 be an integer. One denotes by 𝐸_{𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)} the vector space of formal series Ψ𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)}=_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)}𝜓_{𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!×Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑙_ℎ! (67) that belong to ℂ[[𝑉₀,𝑉₁,(𝑈_ℎ)_{ℎ∈𝐼(𝛼)}]] such that the series Ψ𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}=_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)}𝜓_{𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}exp𝜎𝑟_𝑏(𝛼)𝜌×𝑉^𝑛0₀𝑉^𝑛1₁Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ+𝛼! (68) is convergent. One denotes also by 𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} the vector space of formal series Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0=,𝑊_𝛼≥0Ψ_𝛼𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)}𝑊^𝛼,𝛼! (69) where Ψ_𝛼(𝑉₀,𝑉₁,(𝑈_ℎ)_{ℎ∈𝐼(𝛼)}) belong to 𝐸_{𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)} for all 𝛼≥0, such that the series Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}=_𝛼≥0Ψ_{𝛼𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}𝑊^𝛼 (70) is convergent. One checks that the space 𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} equipped with the norm ‖⋅‖_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} is a Banach space.

In the next two propositions, we study norm estimates for linear operators acting on the Banach spaces 𝐸_{𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)} constructed above.

Proposition 8. Consider a formal series 𝑏𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)}=_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)}𝑏_{𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!×Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑙_ℎ! (71) which is absolutely convergent on the polydisc 𝐷(0,𝑉₀)×𝐷(0,𝑉₁)×_{ℎ∈𝐼(𝛼)}𝐷(0,𝑈_ℎ). One uses the notation |𝑏|𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)}=_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)}𝑏_{𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!×Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑙_ℎ!. (72) Let Ψ(𝑉₀,𝑉₁,(𝑈_ℎ)_{ℎ∈𝐼(𝛼)}) belong to 𝐸_{𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}. Then, the following inequality: 𝑏𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)}×Ψ𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}≤|𝑏|𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)}×Ψ𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)} (73) holds.

Proof. LetΨ𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)}=_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)}𝜓_{𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!×Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑙_ℎ! (74) which belongs to 𝐸_{𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}. By definition, we have that 𝑏𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)}×Ψ𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}=_{𝑛0,𝑛1,𝑙ℎ(𝛼)≥0,ℎ∈𝐼𝑛010,1+𝑛0,2=𝑛,𝑛1,1+𝑛1,2=𝑛𝑙ℎ,1+𝑙ℎ,2=𝑙ℎ(𝛼),ℎ∈𝐼}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!×𝑛_0,1!𝑛_0,2!𝑛_1,1!𝑛_1,2!×Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑙_ℎ,2!⁻¹×𝑏_{𝑛0,1,𝑛1,1,(𝑙ℎ,1)ℎ∈𝐼(𝛼)}×𝜓_{𝑛0,2,𝑛1,2,(𝑙ℎ,2)ℎ∈𝐼(𝛼)}×1exp𝜎𝑟_𝑏(𝛼)𝜌×𝑉^𝑛0₀𝑉^𝑛1₁Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ!.+𝛼 (75) We can give upper bounds for this latter expression 𝑏𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)}Ψ𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}≤∑_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)}∑_{𝑛010,1+𝑛0,2=𝑛,𝑛1,1+𝑛1,2=𝑛𝑙ℎ,1+𝑙ℎ,2=𝑙ℎ,ℎ∈𝐼(𝛼)}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,2!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,2!×𝑛_0,2+𝑛_1,2+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ,2+𝛼!𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ!×𝑏+𝛼_{𝑛0,1,𝑛1,1,(𝑙ℎ,1)ℎ∈𝐼(𝛼)}𝑛_0,1!𝑛_1,1!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑉^𝑛0,1₀𝑉^𝑛1,1₁Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ,1_ℎ×𝜓_{𝑛0,2,𝑛1,2,(𝑙ℎ,2)ℎ∈𝐼(𝛼)}1exp𝜎𝑟_𝑏(𝛼)𝜌𝑉^𝑛0,2₀𝑉^𝑛1,2₁Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ,2_ℎ𝑛_0,2+𝑛_1,2+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ,2!.+𝛼 (76)
Lemma  9. For all integers 𝛼,𝑛₀,𝑛₁≥0, all 𝑙_ℎ≥0, all 0≤𝑛_0,2≤𝑛₀, all 0≤𝑛_1,2≤𝑛₁, and all 0≤𝑙_ℎ,2≤𝑙_ℎ for ℎ∈𝐼(𝛼), one has that𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,2!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,2!𝑛_0,2+𝑛_1,2+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ,2+𝛼!𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ+𝛼!≤1. (77)
Proof . For any integers 𝑎≤𝑏 and 𝛼≥0, one has (𝑎+𝛼)!()!≤𝑏+𝛼𝑎!𝑏! (78) by using the factorization (𝑎+𝛼)!=(𝑎+𝛼)(𝑎+𝛼−1)⋯(𝑎+1)𝑎!. Therefore, one gets the inequality 𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,2!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,2!𝑛_0,2+𝑛_1,2+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ,2+𝛼!𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ!≤𝑛+𝛼₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,2!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,2!𝑛_0,2+𝑛_1,2+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ,2!𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ!. (79) Now, from the identity (𝐴+𝐵)^{𝑛0+𝑛1+∑ℎ∈𝐼(𝛼)𝑙ℎ}=(𝐴+𝐵)^𝑛0(𝐴+𝐵)^𝑛1×Π_{ℎ∈𝐼(𝛼)}(𝐴+𝐵)^𝑙ℎ and the binomial formula, we deduce that 𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,1!𝑛_0,2!𝑛_1,1!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑙_ℎ,2!≤𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,1+𝑛_1,1+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑛_0,2+𝑛_1,2+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ,2! (80) for all 𝑛_0,1+𝑛_0,2=𝑛₀, 𝑛_1,1+𝑛_1,2=𝑛₁, 𝑙_ℎ,1+𝑙_ℎ,2=𝑙_ℎ. Therefore, we deduce that 𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ!𝑛_0,2!𝑛_1,2!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,2!𝑛_0,2+𝑛_1,2+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ,2+𝛼!𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ!≤𝑛+𝛼_0,1!𝑛_1,1!Π_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!𝑛_0,1+𝑛_1,1+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ,1!≤1, (81) and the lemma follows from the inequalities (79) and (81).
Finally, the inequality (73) follows from (76) and (77).

Proposition 10. Let 𝛼,𝛼^′ be integers such that 𝛼^′≥0 and 𝛼^′+1<𝛼. Let 𝑗∈𝐼(𝛼^′) and 𝑘∈{0,1}. One has that 𝜕_𝑈𝑗Ψ𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼′)𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}≤exp−𝜎𝜌𝛼−𝛼^′/(𝛼+1)^𝑏𝑈_𝑗Π^{𝛼−𝛼′−1}_𝑙=1()×Ψ𝑉𝛼−𝑙+1₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼′)𝜌,𝛼′,𝑉0,𝑉1,(𝑈ℎ)′)ℎ∈𝐼(𝛼},𝜕_𝑉𝑘Ψ𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼′)𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}≤exp−𝜎𝜌𝛼−𝛼^′/(𝛼+1)^𝑏𝑉_𝑘Π^{𝛼−𝛼′−1}_𝑙=1()×Ψ𝑉𝛼−𝑙+1₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼′)𝜌,𝛼′,𝑉0,𝑉1,(𝑈ℎ)′)ℎ∈𝐼(𝛼},Ψ𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼′)𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}≤exp−𝜎𝜌𝛼−𝛼^′/(𝛼+1)^𝑏Π^{𝛼−𝛼′}_𝑙=1()×Ψ𝑉𝛼−𝑙+1₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼′)𝜌,𝛼′,𝑉0,𝑉1,(𝑈ℎ)′)ℎ∈𝐼(𝛼} (82)(83)(84) for all Ψ(𝑉₀,𝑉₁,(𝑈_ℎ)_{ℎ∈𝐼(𝛼′)})∈𝐸_{𝜌,𝛼′,𝑉0,𝑉1,(𝑈ℎ)′)ℎ∈𝐼(𝛼}.

Proof . Let Ψ(𝑉₀,𝑉₁,(𝑈_ℎ)_{ℎ∈𝐼(𝛼′)})∈𝐸_{𝜌,𝛼′,𝑉0,𝑉1,(𝑈ℎ)′)ℎ∈𝐼(𝛼} that we write in the form Ψ𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼′)}=_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)}𝜓_{𝑛0,𝑛1,(𝑙ℎ)′)ℎ∈𝐼(𝛼}×Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼′)}𝛿_0,𝑙ℎ𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑙_ℎ!. (85) By definition, we get that 𝜕_𝑈𝑗Ψ𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼′)𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}=_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)}𝜓_{𝑛0,𝑛1,(𝑙ℎ)′ℎ∈𝐼(𝛼),ℎ≠𝑗,𝑙𝑗+1}Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼′)}𝛿_0,𝑙ℎexp𝜎𝑟_𝑏(𝛼)𝜌×𝑉^𝑛0₀𝑉^𝑛1₁Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼)}𝑙_ℎ!.+𝛼 (86) We give upper bounds for this latter expression 𝜕_𝑈𝑗Ψ𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼′)𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}=_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼′)}𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼′),ℎ≠𝑗}𝑙_ℎ+𝑙_𝑗+1+𝛼^′!𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼′)}𝑙_ℎ!×1+𝛼𝑈_𝑗exp𝜎𝜌𝑟_𝑏(𝛼)−𝑟_𝑏𝛼^′×𝜓_{𝑛0,𝑛1,(𝑙ℎ)′ℎ∈𝐼(𝛼),ℎ≠𝑗,𝑙𝑗+1}exp𝜎𝑟_𝑏𝛼^′𝜌×𝑉^𝑛0₀𝑉^𝑛1₁Π_{ℎ∈𝐼(𝛼′),ℎ≠𝑗}𝑈^𝑙ℎ_ℎ𝑈^𝑙𝑗_𝑗⁺¹𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼′),ℎ≠𝑗}𝑙_ℎ+𝑙_𝑗+1+𝛼^′!. (87)
Lemma 11. One has𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼′),ℎ≠𝑗}𝑙_ℎ+𝑙_𝑗+1+𝛼^′!𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼′)}𝑙_ℎ!×1+𝛼exp𝜎𝜌𝑟_𝑏(𝛼)−𝑟_𝑏𝛼^′≤exp−𝜎𝜌𝛼−𝛼^′/(𝛼+1)^𝑏Π^{𝛼−𝛼′−1}_𝑙=1().𝛼−𝑙+1 (88)
Proof. We notice that 𝑟_𝑏(𝛼)−𝑟_𝑏𝛼^′=^𝛼_{𝑛=𝛼′+1}1(𝑛+1)^𝑏≥𝛼−𝛼^′(𝛼+1)^𝑏 (89) and, with the help of (78), that for all integers 𝑎≥0, 𝑎+1+𝛼^′!()!≤1𝑎+𝛼Π^{𝛼−𝛼′−1}_𝑙=1().𝛼−𝑙+1 (90) The lemma follows.
We get that the inequality (82) follows from (87) together with (88). Finally, using similar arguments, one gets also the inequalities (83) and (84).

In the next two propositions, we study norm estimates for linear operators acting on the Banach space 𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}.

Proposition 12. Let a formal series 𝑏(𝑉₀,𝑉₁,𝑈₀,𝑊)∈ℂ[[𝑉₀,𝑉₁,𝑈₀,𝑊]] be absolutely convergent on the polydisc 𝐷(0,𝑉₀)×𝐷(0,𝑉₁)×𝐷(0,𝑈₀)×𝐷(0,𝑊). Let Ψ(𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊) belong to 𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}. Then, the product 𝑏(𝑉₀,𝑉₁,𝑈₀,𝑊)Ψ(𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊) belongs to 𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} and the following inequality: 𝑏𝑉₀,𝑉₁,𝑈₀,𝑊Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤|𝑏|𝑉₀,𝑉₁,𝑈₀,𝑊×Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} (91) holds.

Proof. Let 𝑏𝑉₀,𝑉₁,𝑈₀,𝑊=_𝛼≥0𝑏_𝛼𝑉₀,𝑉₁,𝑈₀𝑊^𝛼,Ψ𝑉𝛼!₀,𝑉₁,𝑈_ℎℎ≥0=,𝑊_𝛼≥0Ψ_𝛼𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼)}𝑊^𝛼.𝛼! (92) By definition, we get 𝑏𝑉₀,𝑉₁,𝑈₀,𝑊Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}=_{𝛼≥0𝛼1+𝛼2=𝛼}𝑏𝛼!^𝛼1𝑉₀,𝑉₁,𝑈₀𝛼₁!×Ψ_𝛼2𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2)}𝛼₂!_{𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}𝑊^𝛼. (93)
Lemma 13. One has𝑏_𝛼1𝑉₀,𝑉₁,𝑈₀Ψ_𝛼2𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2)𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}≤𝛼₂!𝑏𝛼!^𝛼1𝑉₀,𝑉₁,𝑈₀×Ψ_𝛼2𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2)𝜌,𝛼2,𝑉0,𝑉1,(𝑈ℎ)2)ℎ∈𝐼(𝛼}. (94)
Proof. We can write 𝑏_𝛼1𝑉₀,𝑉₁,𝑈₀=_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)}𝑏_{𝛼1,𝑛0,𝑛1,𝑙0}×Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑙_ℎ!,Ψ_𝛼2𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2)}=_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼)}𝜓_{𝛼2,𝑛0,𝑛1,(𝑙ℎ)2)ℎ∈𝐼(𝛼}Π_{ℎ∈𝐼(𝛼)⧵𝐼(𝛼2)}×𝛿_0,𝑙ℎ𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑙_ℎ!. (95) By remembering (73) of Proposition 8, we deduce that 𝑏_𝛼1𝑉₀,𝑉₁,𝑈₀Ψ_𝛼2𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2)𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}≤𝑏_𝛼1𝑉₀,𝑉₁,𝑈₀×_{𝑛0,𝑛1,𝑙ℎ𝛼≥0,ℎ∈𝐼2}𝜓_{𝛼2,𝑛0,𝑛1,(𝑙ℎ)2)ℎ∈𝐼(𝛼}exp𝜎𝑟_𝑏(𝛼)𝜌×𝑉^𝑛0₀𝑉^𝑛1₁Π_{ℎ∈𝐼(𝛼2)}𝑈^𝑙ℎ_ℎ𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼2)}𝑙_ℎ!.+𝛼 (96)
Lemma 14. One has1𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼2)}𝑙_ℎ!≤𝛼+𝛼₂!1𝛼!𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼2)}𝑙_ℎ+𝛼₂!. (97)
Proof. We write 1𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼2)}𝑙_ℎ!=𝑛+𝛼₀+𝑛₁+∑_{ℎ∈𝐼(𝛼2)}𝑙_ℎ+𝛼₂!𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼2)}𝑙_ℎ!1+𝛼𝑛₀+𝑛₁+∑_{ℎ∈𝐼(𝛼2)}𝑙_ℎ+𝛼₂! (98) and we use the inequality 𝑎+𝛼₂!()!≤𝛼𝑎+𝛼²!𝛼! (99) for all 𝛼=𝛼₁+𝛼₂ and all 𝑎∈ℕ which follows from (78). This yields the lemma.
Using the fact that exp(𝜎𝑟_𝑏(𝛼)𝜌)≥exp(𝜎𝑟_𝑏(𝛼₂)𝜌) and gathering the inequalities (96) and (97) yield (94).
Finally, using (93) with (94), one gets 𝑏𝑉₀,𝑉₁,𝑈₀𝑉,𝑊×Ψ₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤_{𝛼≥0𝛼1+𝛼2=𝛼}𝑏_𝛼1𝑉₀,𝑉₁,𝑈₀𝛼₁!×Ψ_𝛼2𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2)𝜌,𝛼2,𝑉0,𝑉1,(𝑈ℎ)2)ℎ∈𝐼(𝛼}×𝑊^𝛼 (100) from which the inequality (91) follows.

Proposition 15. (1) Let 𝑆,𝑘≥0 be integers such that 𝑆≥𝑘+1+max𝑏𝑑_1,𝑘+2+3,𝑑+1+𝑏𝑑+𝑑_1,𝑘+1. (101) Then, there exists a constant 𝐶_8.1>0 (which is independent of 𝜌>1) such that 𝐵_1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘×𝔻_𝐀Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤𝐶_8.1𝑊^𝑆−𝑘×Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} (102) for all Ψ(𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊)∈𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}.
(2) Let 𝑆,𝑘≥0 be integers such that 𝑆≥𝑘+3+𝑏2+𝑑_2,𝑘. (103) Then, there exists a constant 𝐶_8.2>0 (which is independent of 𝜌>1) such that 𝐵_2,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘×𝔻_𝐁Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤𝐶_8.2𝑊^𝑆−𝑘×Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} (104) for all Ψ(𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊)∈𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}.

Proof. (1) We show the first inequality (102). We expand 𝐵_1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊=_𝛼≥0𝐵_1,𝑘,𝛼𝑉₀,𝑉₁,𝑈₀𝑊^𝛼.𝛼! (105) By definition, we have 𝐵_1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝔻_𝐀Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}=∑_𝛼≥0∑_{𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝐵𝛼!^1,𝑘,𝛼1𝑉₀,𝑉₁,𝑈₀𝛼₁!×∑_{𝑗∈𝐼(𝛼2−𝑆+𝑘)}𝐀_{𝑗,𝛼2−𝑆+𝑘+1}𝑉₀,𝑉₁,(𝑈_ℎ)_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}𝛼₂!×𝜕_𝑈𝑗Ψ_{𝛼2−𝑆+𝑘}𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}𝑊^𝛼. (106) Now, using Lemma 13, we deduce that𝐵_1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝔻_𝐀Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤_{𝛼≥0𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝐵_1,𝑘,𝛼1𝑉₀,𝑉₁,𝑈₀𝛼₁!×_{𝛼𝑗∈𝐼2−𝑆+𝑘}𝐀_{𝑗,𝛼2−𝑆+𝑘+1}𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘+1)}×𝜕_𝑈𝑗Ψ_{𝛼2−𝑆+𝑘}𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2,𝑉0,𝑉1,(𝑈ℎ)2)ℎ∈𝐼(𝛼}𝑊^𝛼. (107)In the next lemma, we give estimates for the coefficients of the series 𝐀_𝑗,𝛼 and |𝐵_1,𝑘,𝛼|.
Lemma 16. (1)  The coefficients of the Taylor series of 𝐀_{𝑗,𝛼2−𝑆+𝑘+1}(𝑉₀,𝑉₁,(𝑈_ℎ)_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)})𝐀_{𝑗,𝛼2−𝑆+𝑘+1}𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘+1)}=_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}𝐴_{𝑗,𝛼2−𝑆+𝑘+1,𝑛0,𝑛1,(𝑙ℎ)2ℎ∈𝐼(𝛼−𝑆+𝑘+1)}×𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!Π_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}𝑈^𝑙ℎ_ℎ𝑙_ℎ! (108)satisfy the next estimates. There exist constants 𝑎,𝛿>0, with 𝛿>𝛿, 𝑎_𝑝>0, 0≤𝑝≤𝑑  such that𝐴_{𝑗,𝛼2−𝑆+𝑘+1,𝑛0,𝑛1,(𝑙ℎ)2ℎ∈𝐼(𝛼−𝑆+𝑘+1)}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}𝑙_ℎ!≤𝑎𝜈𝛼₂−𝑆+𝑘+1²+()𝜌+𝛿𝑑+1max_{0≤𝑝≤𝑑}𝑎_𝑝𝜌+𝛿^𝑑𝒫_𝑑𝛼₂×𝛿−𝑆+𝑘_{𝑛0+𝑛1+∑2ℎ∈𝐼(𝛼−𝑆+𝑘+1)𝑙ℎ−1} (109)for all 𝛼₂≥𝑆−𝑘, all 𝑗∈𝐼(𝛼₂−𝑆+𝑘), all 𝑛₀,𝑛₁,𝑙_ℎ≥0, ℎ∈𝐼(𝛼₂−𝑆+𝑘+1)  where 𝒫_𝑑  is defined in (115).
(2)  The coefficients of the Taylor series of |𝐵_1,𝑘,𝛼1|(𝑉₀,𝑉₁,𝑈₀)𝐵_1,𝑘,𝛼1𝑉₀,𝑉₁,𝑈₀=_{𝑛0,𝑛1,𝑙0≥0}𝑏_{1,𝑘,𝛼1,𝑛0,𝑛1,𝑙0}𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!𝑈^𝑙0₀𝑙₀! (110)satisfy the following inequalities. There exist constants 𝛿>𝛿, 𝐷_1,𝑘,𝐷_1,𝑘>0  with 𝑏_{1,𝑘,𝛼1,𝑛0,𝑛1,𝑙0}𝑛₀!𝑛₁!𝑙₀!≤𝐷_1,𝑘𝜌+𝛿^𝑑1,𝑘𝛼₁!𝐷^𝛼1_1,𝑘𝛿^{𝑛0+𝑛1+𝑙0} (111)for all 𝛼₁≥0, all 𝑛₀,𝑛₁,𝑙₀≥0.
Proof. We first treat the estimates for 𝐀_𝑗,𝛼. From the Cauchy formula in several variables, one can write 𝜕^𝑛0_𝑣0𝜕^𝑛1_𝑣1Π_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}𝜕^𝑙ℎ_𝑢ℎ𝐴_𝑗𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘+1)}×𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}𝑙_ℎ!⁻¹=12𝑖𝜋^{𝛼2−𝑆+𝑘+4}×_{𝐶(𝑣0,𝛿)𝐶(𝑣1,𝛿)}Π_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}×_{𝐶(𝑢ℎ,𝛿)}𝐴_𝑗𝜒₀,𝜒₁,𝜉_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘+1)}×𝑑𝜒₀𝑑𝜒₁Π_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}𝑑𝜉_ℎ×𝜒₀−𝑣₀^𝑛0+1𝜒₁−𝑣₁^𝑛1+1×Π_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}𝜉_ℎ−𝑢_ℎ^{𝑙ℎ+1−1} (112) for all |𝑣₀|<𝑅, |𝑣₁|<𝑅, |𝑢_ℎ|<𝜌, ℎ∈𝐼(𝛼₂−𝑆+𝑘+1) and 𝑗∈𝐼(𝛼₂−𝑆+𝑘) where 𝑅 is introduced in Section 2.2. The integration is made along positively oriented circles with radius 𝛿>0, 𝐶(𝑣₀,𝛿),𝐶(𝑣₁,𝛿) and 𝐶(𝑢_ℎ,𝛿) for ℎ∈𝐼(𝛼₂−𝑆+𝑘+1). We choose the real number 𝛿>𝛿 in such a way that 𝑅+𝛿<𝑅^′ where 𝑅^′ is defined in Section 2.1 and 𝛿 at the beginning of Section 3.1. Now, since the functions 𝑎(𝜒₀,𝜒₁) and 𝑎_𝑝(𝜒₀,𝜒₁) are holomorphic on 𝐷(0,𝑅^′)², the number 𝜈>0 (see (10)) can be chosen large enough such that there exist real numbers 𝑎,𝑎_𝑝>0, for 0≤𝑝≤𝑑, with sup_{|𝜒0|<𝑅+𝛿,|𝜒1|<𝑅+𝛿}𝜕^𝑙1_𝜒1𝑎𝜒₀,𝜒₁𝑙₁!𝜈^𝑙1≤𝑎,sup_{|𝜒0|<𝑅+𝛿,|𝜒1|<𝑅+𝛿}𝜕^𝑙0_𝜒1𝑎_𝑝𝜒₀,𝜒₁𝑙₀!𝜈^𝑙0≤𝑎_𝑝 (113) for all 𝑙₀,𝑙₁≥0. We recall also that for any integers 𝑘,𝑛≥1, the number of tuples (𝑏₁,…,𝑏_𝑘)∈ℕ^𝑘 such that 𝑏₁+⋯+𝑏_𝑘=𝑛 is (𝑛+𝑘−1)!/((𝑘−1)!𝑛!). From these latter statements and the definition of 𝐴_𝑗 given by (27), we deduce that 𝐴_𝑗𝜒₀,𝜒₁,𝜉_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘+1)}≤𝑎𝜈𝑗+1²+()𝜌+𝛿𝑑+1max_{0≤𝑝≤𝑑}𝑎_𝑝𝜌+𝛿^𝑑𝒫_𝑑𝑗 (114) (since 𝜌>1), where 𝒫_𝑑𝑗=𝑗+𝑑!𝑗!=Π^𝑑_𝑙=1𝑗+𝑙 (115) is a polynomial of degree 𝑑 in 𝑗 with positive coefficients, for all |𝜒₀|<𝑅+𝛿, |𝜒₁|<𝑅+𝛿, |𝜉_ℎ|<𝜌+𝛿, ℎ∈𝐼(𝛼₂−𝑆+𝑘+1) and 𝑗∈𝐼(𝛼₂−𝑆+𝑘). Gathering (112) and (114) yields (109).
Again, from the Cauchy formula in several variables, one can write 𝜕^𝑛0_𝑣0𝜕^𝑛1_𝑣1𝜕^𝑙0_𝑢0𝑏_1,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀𝑛₀!𝑛₁!𝑙₀!=12𝑖𝜋³×_{𝐶(𝑣0,𝛿)𝐶(𝑣1,𝛿)𝐶(𝑢0,𝛿)}𝑏_1,𝑘,𝛼1𝜒₀,𝜒₁,𝜉₀×𝑑𝜒₀𝑑𝜒₁𝑑𝜉₀×𝜒₀−𝑣₀^𝑛0+1𝜒₁−𝑣₁^𝑛1+1×(𝜉₀−𝑢₀)^𝑙0+1−1 (116) for all |𝑣₀|<𝑅, |𝑣₁|<𝑅, |𝑢₀|<𝜌. Again, one chooses the real number 𝛿>𝛿 in such a way that 𝑅+𝛿<𝑅^′. By construction of 𝑏_1,𝑘,𝛼1 in Section 2.2, we know that there exist two constants 𝐷_1,𝑘,𝐷_1,𝑘>0 such that 𝑏_1,𝑘,𝛼1𝜒₀,𝜒₁,𝜉₀≤𝐷_1,𝑘𝜌+𝛿^𝑑1,𝑘𝛼₁!𝐷^𝛼1_1,𝑘 (117) for all 𝛼₁≥0, all |𝜒₀|<𝑅+𝛿, |𝜒₁|<𝑅+𝛿, |𝜉₀|<𝜌+𝛿. Gathering (116) and (117) yields (111).
From (111), we deduce that 𝐵_1,𝑘,𝛼1𝑉₀,𝑉₁,𝑈₀𝛼₁!≤𝐷_1,𝑘𝜌+𝛿^𝑑1,𝑘𝐷^𝛼1_1,𝑘1−𝑉₀/𝛿1−𝑉₁/𝛿1−𝑈₀./𝛿 (118) On the other hand, from Proposition 8, we deduce that 𝐀_{𝑗,𝛼2−𝑆+𝑘+1}𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘+1)}×𝜕_𝑈𝑗Ψ_{𝛼2−𝑆+𝑘}×𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2,𝑉0,𝑉1,(𝑈ℎ)2)ℎ∈𝐼(𝛼}≤𝐀_{𝑗,𝛼2−𝑆+𝑘+1}𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘+1)}×𝜕_𝑈𝑗Ψ_{𝛼2−𝑆+𝑘}×𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2,𝑉0,𝑉1,(𝑈ℎ)2)ℎ∈𝐼(𝛼}. (119) From (109), we deduce that 𝐀_{𝑗,𝛼2−𝑆+𝑘+1}𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘+1)}≤𝑎𝜈𝛼₂−𝑆+𝑘+1²𝜌+𝛿+(𝑑+1)×max_{0≤𝑝≤𝑑}𝑎_𝑝𝜌+𝛿^𝑑𝒫_𝑑𝛼₂×−𝑆+𝑘1−𝑉₀𝛿1−𝑉₁𝛿×Π_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}1−𝑈_ℎ𝛿⁻¹ (120) for all 𝑗∈𝐼(𝛼₂−𝑆+𝑘). Now, from the definition of 𝑈_ℎ=𝛿/(ℎ^𝑏+1), where 𝑏>1, we know that there exists 𝜅>0 such that Π_{ℎ∈𝐼(𝛼)}1−𝑈_ℎ𝛿≥𝜅 (121) for all 𝛼≥0. From Proposition 10, we have that 𝜕_𝑈𝑗Ψ_{𝛼2−𝑆+𝑘}𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2,𝑉0,𝑉1,(𝑈ℎ)2)ℎ∈𝐼(𝛼}≤exp−𝜎𝜌(𝑆−𝑘)/𝛼₂+1^𝑏𝑈_𝑗Π^{𝑆−𝑘−1}_𝑙=1𝛼₂×Ψ−𝑙+1_{𝛼2−𝑆+𝑘}×𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2−𝑆+𝑘,𝑉0,𝑉1,(𝑈ℎ)2ℎ∈𝐼(𝛼−𝑆+𝑘)}. (122) Collecting the estimates (120), (121), and (122), we get from (119) that _{𝛼𝑗∈𝐼2−𝑆+𝑘}𝐀_{𝑗,𝛼2−𝑆+𝑘+1}𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘+1)}×𝜕_𝑈𝑗Ψ_{𝛼2−𝑆+𝑘}×𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2,𝑉0,𝑉1,(𝑈ℎ)2)ℎ∈𝐼(𝛼}≤𝒜_𝜌,𝛼2×Ψ_{𝛼2−𝑆+𝑘}×𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2−𝑆+𝑘,𝑉0,𝑉1,(𝑈ℎ)2ℎ∈𝐼(𝛼−𝑆+𝑘)}, (123) where 𝒜_𝜌,𝛼2=𝑎𝜈𝛼₂−𝑆+𝑘+1²𝜌+𝛿+(𝑑+1)×max_{0≤𝑝≤𝑑}𝑎_𝑝𝜌+𝛿^𝑑𝒫_𝑑𝛼₂×−𝑆+𝑘1−𝑉₀𝛿1−𝑉₁𝛿𝜅⁻¹×exp−𝜎𝜌(𝑆−𝑘)/𝛼₂+1^𝑏Π^{𝑆−𝑘−1}_𝑙=1𝛼₂1−𝑙+1𝛿×𝛼₂×𝛼−𝑆+𝑘+1₂−𝑆+𝑘^𝑏+1. (124) Now, we recall the following classical estimates. Let 𝛿,𝑚₁,𝑚₂>0 be positive real numbers, and then sup_𝑥≥0(𝑥+𝛿)^𝑚1exp−𝑚₂𝑥≤𝑚₁𝑚₂^𝑚1exp−𝑚₁exp𝛿𝑚₂ (125) holds. Hence, (𝜌+𝛿)^𝑑1,𝑘𝒜_𝜌,𝛼2≤𝑎𝜈𝛼₂−𝑆+𝑘+1²𝛼₂+1^{𝑏(1+𝑑1,𝑘)}×exp(−1)1+𝑑_1,𝑘𝜎(𝑆−𝑘)^1+𝑑1,𝑘((×exp𝛿𝜎𝑆−𝑘))1−𝑉₀𝛿1−𝑉₁𝛿𝜅⁻¹+(𝑑+1)max_{0≤𝑝≤𝑑}𝑎_𝑝𝒫_𝑑𝛼₂−𝑆+𝑘𝛼₂+1^{𝑏(𝑑+𝑑1,𝑘)}×𝑑+𝑑_1,𝑘exp(−1)𝜎(𝑆−𝑘)^{𝑑+𝑑1,𝑘}((×exp𝛿𝜎𝑆−𝑘))1−𝑉₀𝛿1−𝑉₁𝛿𝜅⁻¹×𝛼₂−𝑆+𝑘+1𝛼₂−𝑆+𝑘^𝑏+1𝛿Π^{𝑆−𝑘−1}_𝑙=1𝛼₂.−𝑙+1 (126) Under the assumptions (101), one gets a constant 𝐶_8.1>0 (depending on 𝑎,max_{0≤𝑝≤𝑑}𝑎_𝑝,𝛿,𝛿,𝑏,𝑑,𝑑_1,𝑘,𝜎,𝜈,𝑆,𝑘,𝜅,𝑉₀,𝑉₁) such that 𝜌+𝛿^𝑑1,𝑘𝒜_𝜌,𝛼2≤𝐶_8.1 (127) for all 𝜌≥0, all 𝛼₂≥𝑆−𝑘. Finally, gathering (107), (118), (123), and (127), one gets that 𝐵_1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝔻_𝐀Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤∑_𝛼≥0∑_{𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝐷_1,𝑘1−𝑉₀/𝛿1−𝑉₁/𝛿1−𝑈₀𝐷/𝛿^𝛼11,𝑘×𝐶_8.1Ψ_{𝛼2−𝑆+𝑘}𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2−𝑆+𝑘,𝑉0,𝑉1,(𝑈ℎ)2ℎ∈𝐼(𝛼−𝑆+𝑘)}×𝑊^{𝛼1+𝛼2−𝑆+𝑘}𝑊^𝑆−𝑘=𝐶_8.1𝐷_1,𝑘1−𝑉₀/𝛿1−𝑉₁/𝛿1−𝑈₀/𝛿1−𝐷_1,𝑘𝑊×𝑊^𝑆−𝑘Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} (128)
provided that 𝑉₀<𝛿, 𝑉₁<𝛿, 𝑈₀<𝛿, and 𝑊<1/𝐷_1,𝑘, which yields (102).
(2) Now, we turn towards the estimates (104) which will follow from the same arguments as in (1). Using Lemma 13, we get that 𝐵_2,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘×𝔻_𝐁Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤_{𝛼≥0𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝐵_2,𝑘,𝛼1𝑉₀,𝑉₁,𝑈₀𝛼₁!×_{𝛼𝑗∈𝐼2−𝑆+𝑘}𝐁_{𝑗,𝛼2−𝑆+𝑘+1}𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘+1)}×𝜕_𝑈𝑗Ψ_{𝛼2−𝑆+𝑘}×𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2,𝑉0,𝑉1,(𝑈ℎ)2)ℎ∈𝐼(𝛼}×𝑊^𝛼. (129) In the next lemma, we give estimates for the coefficients of the series 𝐁_𝑗,𝛼 and |𝐵_2,𝑘,𝛼|.
Lemma 17. (1) The coefficients of the Taylor series of 𝐁_{𝑗,𝛼2−𝑆+𝑘+1}(𝑉₀,𝑉₁,(𝑈_ℎ)_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)})𝐁_{𝑗,𝛼2−𝑆+𝑘+1}𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘+1)}=_{𝑛0,𝑛1,𝑙ℎ≥0,ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}𝐵_{𝑗,𝛼2−𝑆+𝑘+1,𝑛0,𝑛1,(𝑙ℎ)2ℎ∈𝐼(𝛼−𝑆+𝑘+1)}×𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!Π_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}𝑈^𝑙ℎ_ℎ𝑙_ℎ! (130)satisfy the next estimates. There exist a constant 𝛿>0, with 𝛿>𝛿  such that𝐵_{𝑗,𝛼2−𝑆+𝑘+1,𝑛0,𝑛1,(𝑙ℎ)2ℎ∈𝐼(𝛼−𝑆+𝑘+1)}𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}𝑙_ℎ!≤𝜈𝛼₂−𝑆+𝑘+1𝜌+𝛿𝛿^{𝑛0+𝑛1+∑2ℎ∈𝐼(𝛼−𝑆+𝑘+1)𝑙ℎ} (131)for all 𝛼₂≥𝑆−𝑘, all 𝑗∈𝐼(𝛼₂−𝑆+𝑘), all 𝑛₀,𝑛₁,𝑙_ℎ≥0, ℎ∈𝐼(𝛼₂−𝑆+𝑘+1).
(2)The coefficients of the Taylor series of |𝐵_2,𝑘,𝛼1|(𝑉₀,𝑉₁,𝑈₀)𝐵_2,𝑘,𝛼1𝑉₀,𝑉₁,𝑈₀=_{𝑛0,𝑛1,𝑙0≥0}𝑏_{2,𝑘,𝛼1,𝑛0,𝑛1,𝑙0}𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!𝑈^𝑙0₀𝑙₀! (132)satisfy the following inequalities. There exist constants 𝛿>𝛿, 𝐷_2,𝑘,𝐷_2,𝑘>0 with𝑏_{2,𝑘,𝛼1,𝑛0,𝑛1,𝑙0}𝑛₀!𝑛₁!𝑙₀!≤𝐷_2,𝑘𝜌+𝛿^𝑑2,𝑘𝛼₁!𝐷^𝛼1_2,𝑘𝛿^{𝑛0+𝑛1+𝑙0} (133)for all 𝛼₁≥0, all 𝑛₀,𝑛₁,𝑙₀≥0.
Proof. (1) From the Cauchy formula in several variables, one can check that 𝜕^𝑛0_𝑣0𝜕^𝑛1_𝑣1Π_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}𝜕^𝑙ℎ_𝑢ℎ×𝐵_𝑗𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘+1)}×𝑛₀!𝑛₁!Π_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}𝑙_ℎ!⁻¹=12𝑖𝜋^{𝛼2−𝑆+𝑘+4}×_{𝐶(𝑣0,𝛿)𝐶(𝑣1,𝛿)}Π_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}×_{𝐶(𝑢ℎ,𝛿)}𝐵_𝑗𝜒₀,𝜒₁,𝜉_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘+1)}×𝑑𝜒₀𝑑𝜒₁Π_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}𝑑𝜉_ℎ×(𝜒₀−𝑣₀)^𝑛0+1(𝜒₁−𝑣₁)^𝑛1+1×Π_{ℎ∈𝐼(𝛼2−𝑆+𝑘+1)}(𝜉_ℎ−𝑢_ℎ)^{𝑙ℎ+1−1} (134) for all |𝑣₀|<𝑅, |𝑣₁|<𝑅, |𝑢_ℎ|<𝜌, ℎ∈𝐼(𝛼₂−𝑆+𝑘+1), and 𝑗∈𝐼(𝛼₂−𝑆+𝑘). We choose the real number 𝛿>𝛿 in such a way that 𝑅+𝛿<𝑅^′. From the definition given in (28), we get that 𝐵_𝑗𝜒₀,𝜒₁,𝜉_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘+1)}≤𝜈𝑗+1𝜌+𝛿 (135) for all |𝜒₀|<𝑅+𝛿, |𝜒₁|<𝑅+𝛿, |𝜉_ℎ|<𝜌+𝛿, ℎ∈𝐼(𝛼₂−𝑆+𝑘+1), and 𝑗∈𝐼(𝛼₂−𝑆+𝑘). Gathering (134) and (135) yields (131).
(2) The proof is exactly the same as (2) in Lemma 16.
From (133), we deduce that 𝐵_2,𝑘,𝛼1𝑉₀,𝑉₁,𝑈₀𝛼₁!≤𝐷_2,𝑘𝜌+𝛿^𝑑2,𝑘𝐷^𝛼1_2,𝑘1−𝑉₀/𝛿1−𝑉₁/𝛿1−𝑈₀./𝛿 (136) Using Propositions 8 and 10, we deduce that _{𝛼𝑗∈𝐼2−𝑆+𝑘}𝐁_{𝑗,𝛼2−𝑆+𝑘+1}𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘+1)}×𝜕_𝑈𝑗Ψ_{𝛼2−𝑆+𝑘}×𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+k)𝜌,𝛼2,𝑉0,𝑉1,(𝑈ℎ)2)ℎ∈𝐼(𝛼}≤ℬ_𝜌,𝛼2×Ψ_{𝛼2−𝑆+𝑘}×𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2−𝑆+𝑘,𝑉0,𝑉1,(𝑈ℎ)2ℎ∈𝐼(𝛼−𝑆+𝑘)}, (137) where ℬ_𝜌,𝛼2=𝜈𝛼₂−𝑆+𝑘+1𝜌+𝛿1−𝑉₀/𝛿1−𝑉₁𝜅×()/𝛼/𝛿exp−𝜎𝜌𝑆−𝑘₂+1^𝑏Π^{𝑆−𝑘−1}_𝑙=1𝛼₂×1−𝑙+1𝛿𝛼₂−𝑆+𝑘+1𝛼₂−𝑆+𝑘^𝑏+1 (138) and where 𝜅 is introduced in (121). Using the estimates (125), we get (𝜌+𝛿)^𝑑2,𝑘ℬ_𝜌,𝛼2≤𝜈𝛼₂−𝑆+𝑘+1𝛼₂+1^{𝑏(1+𝑑2,𝑘)}×exp(−1)1+𝑑_2,𝑘𝜎(𝑆−𝑘)^1+𝑑2,𝑘((×exp𝛿𝜎𝑆−𝑘))1−𝑉₀𝛿1−𝑉₁𝛿𝜅⁻¹×𝛼₂−𝑆+𝑘+1𝛼₂−𝑆+𝑘^𝑏+1𝛿Π^{𝑆−𝑘−1}_𝑙=1𝛼₂.−𝑙+1 (139) Under the assumptions (103), one gets a constant 𝐶_8.2>0 (depending on 𝛿,𝛿,𝑏,𝑑_2,𝑘,𝜎,𝜈,𝑆,𝑘,𝜅,𝑉₀,𝑉₁) such that 𝜌+𝛿^𝑑2,𝑘ℬ_𝜌,𝛼2≤𝐶_8.2 (140) for all 𝜌≥0, all 𝛼₂≥𝑆−𝑘. Finally, gathering (129), (136), (137), and (140), we get (104).

Proposition 18. (1)  Let 𝑆,𝑘≥0 be integers such that 𝑆≥𝑘+1+𝑏max𝑑_1,𝑘,𝑑_2,𝑘. (141) Then, for 𝑚∈{0,1}, there exists a constant 𝐶₉>0 (which is independent of 𝜌>1) such that 𝐵_𝑚+1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝜕_𝑉𝑚×Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤𝐶₉𝑊^𝑆−𝑘×Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} (142) for all Ψ(𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊)∈𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}.
(2) Let 𝑆,𝑘≥0 be integers such that 𝑆≥𝑘+𝑏𝑑_3,𝑘. (143) Then, there exists a constant 𝐶_9.1>0 (which is independent of 𝜌>1) such that 𝐵_3,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘×Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤𝐶_9.1𝑊^𝑆−𝑘×Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} (144) for all Ψ(𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊)∈𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}.

Proof. (1) We expand 𝐵_𝑚+1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊=_𝛼≥0𝐵_{𝑚+1,𝑘,𝛼}𝑉₀,𝑉₁,𝑈₀𝑊^𝛼.𝛼! (145) By definition, we have 𝐵_𝑚+1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝜕_𝑉𝑚×Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}=_{𝛼≥0𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝐵𝛼!^{𝑚+1,𝑘,𝛼1}𝑉₀,𝑉₁,𝑈₀𝛼₁!×𝜕_𝑉𝑚Ψ_{𝛼2−𝑆+𝑘}×𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘}×𝛼₂!⁻¹_{𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}×𝑊^𝛼. (146) Now, using Lemma 13, we deduce that 𝐵_𝑚+1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝜕_𝑉𝑚×Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤_{𝛼≥0𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝐵_{𝑚+1,𝑘,𝛼1}𝑉₀,𝑉₁,𝑈₀𝛼₁!×𝜕_𝑉𝑚Ψ_{𝛼2−𝑆+𝑘}×𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2,𝑉0,𝑉1,(𝑈ℎ)2)ℎ∈𝐼(𝛼}×𝑊^𝛼. (147) From Proposition 10, we know that 𝜕_𝑉𝑚Ψ_{𝛼2−𝑆+𝑘}×𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2,𝑉0,𝑉1,(𝑈ℎ)2)ℎ∈𝐼(𝛼}≤exp−𝜎𝜌(𝑆−𝑘)/𝛼₂+1^𝑏𝑉_𝑚Π^{𝑆−𝑘−1}_𝑙=1𝛼₂×Ψ−𝑙+1_{𝛼2−𝑆+𝑘}×𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2−𝑆+𝑘,𝑉0,𝑉1,(𝑈ℎ)2ℎ∈𝐼(𝛼−𝑆+𝑘)}. (148) From (118), (136), (147), and (148), we get that 𝐵_𝑚+1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝜕_𝑉𝑚×Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤_{𝛼≥0𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝐷^𝛼1_𝑚+1,𝑘×𝒞_𝜌,𝛼2×Ψ_{𝛼2−𝑆+𝑘}×𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2−𝑆+𝑘,𝑉0,𝑉1,(𝑈ℎ)2ℎ∈𝐼(𝛼−𝑆+𝑘)}×𝑊^𝛼, (149) where 𝒞_𝜌,𝛼2=𝐷_𝑚+1,𝑘𝜌+𝛿^{𝑑𝑚+1,𝑘}1−𝑉₀/𝛿1−𝑉₁/𝛿1−𝑈₀×()/𝛼/𝛿exp−𝜎𝜌𝑆−𝑘₂+1^𝑏𝑉_𝑚Π^{𝑆−𝑘−1}_𝑙=1𝛼₂.−𝑙+1 (150)
Using the estimates (125), we deduce that 𝒞_𝜌,𝛼2≤𝐷_𝑚+1,𝑘𝑑_𝑚+1,𝑘exp(−1)/𝜎(𝑆−𝑘)^{𝑑𝑚+1,𝑘}exp(𝛿𝜎(𝑆−𝑘))1−𝑉₀/𝛿1−𝑉₁/𝛿1−𝑈₀×𝛼/𝛿₂+1^{𝑏𝑑𝑚+1,𝑘}𝑉_𝑚Π^{𝑆−𝑘−1}_𝑙=1𝛼₂.−𝑙+1 (151)
Under the assumption (141), we get a constant 𝐶₉>0 (depending on 𝐷_𝑚+1,𝑘,𝑑_𝑚+1,𝑘,𝑆,𝑘,𝛿,𝜎,𝑉₀,𝑉₁,𝑈₀,𝑏) such that 𝒞_𝜌,𝛼2≤𝐶₉ (152) for all 𝜌>1, all 𝛼₂≥𝑆−𝑘. Finally, collecting (149) and (152), we get 𝐵_𝑚+1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝜕_𝑉𝑚Ψ(𝑉₀,𝑉₁,𝑈_ℎ)_ℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤∑_𝛼≥0∑_{𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝐷^𝛼1_𝑚+1,𝑘×𝐶₉Ψ_{𝛼2−𝑆+𝑘}𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2−𝑆+𝑘,𝑉0,𝑉1,(𝑈ℎ)2ℎ∈𝐼(𝛼−𝑆+𝑘)}𝑊^{𝛼1+𝛼2−𝑆+𝑘}𝑊^𝑆−𝑘=𝐶₉1−𝐷_𝑚+1,𝑘𝑊𝑊^𝑆−𝑘Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} (153) which yields (142).
(2) We expand 𝐵_3,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊=_𝛼≥0𝐵_3,𝑘,𝛼𝑉₀,𝑉₁,𝑈₀𝑊^𝛼.𝛼! (154)
By definition, we have 𝐵_3,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘×Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}=_{𝛼≥0𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝐵𝛼!^3,𝑘,𝛼1𝑉₀,𝑉₁,𝑈₀𝛼₁!×Ψ_{𝛼2−𝑆+𝑘}𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘}×𝛼₂!⁻¹_{𝜌,𝛼,𝑉0,𝑉1,(𝑈ℎ)ℎ∈𝐼(𝛼)}×𝑊^𝛼. (155) Now, using Lemma 13, we deduce that 𝐵_3,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘×Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}=_{𝛼≥0𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝐵_3,𝑘,𝛼1𝑉₀,𝑉₁,𝑈₀𝛼₁!×Ψ_{𝛼2−𝑆+𝑘}×𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2,𝑉0,𝑉1,(𝑈ℎ)2)ℎ∈𝐼(𝛼}×𝑊^𝛼. (156) From Proposition 10, we know that Ψ_{𝛼2−𝑆+𝑘}(𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2,𝑉0,𝑉1,(𝑈ℎ)2)ℎ∈𝐼(𝛼}≤exp−𝜎𝜌(𝑆−𝑘)/𝛼₂+1^𝑏Π^𝑆−𝑘_𝑙=1𝛼₂×Ψ−𝑙+1_{𝛼2−𝑆+𝑘}×𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2−𝑆+𝑘,𝑉0,𝑉1,(𝑈ℎ)2ℎ∈𝐼(𝛼−𝑆+𝑘)}. (157) On the other hand, the coefficients of the Taylor series of |𝐵_3,𝑘,𝛼1|(𝑉₀,𝑉₁,𝑈₀)𝐵_3,𝑘,𝛼1𝑉₀,𝑉₁,𝑈₀=_{𝑛0,𝑛1,𝑙0≥0}𝑏_{3,𝑘,𝛼1,𝑛0,𝑛1,𝑙0}𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!𝑈^𝑙0₀𝑙₀! (158) satisfy the following inequalities. There exist constants 𝛿>𝛿, 𝐷_3,𝑘,𝐷_3,𝑘>0 with 𝑏_{3,𝑘,𝛼1,𝑛0,𝑛1,𝑙0}𝑛₀!𝑛₁!𝑙₀!≤𝐷_3,𝑘𝜌+𝛿^𝑑3,𝑘𝛼₁!𝐷^𝛼1_3,𝑘𝛿^{𝑛0+𝑛1+𝑙0} (159) for all 𝛼₁≥0, all 𝑛₀,𝑛₁,𝑙₀≥0. The proof copies (2) from Lemma 16. From (159), we deduce that 𝐵_3,𝑘,𝛼1𝑉₀,𝑉₁,𝑈₀𝛼₁!≤𝐷_3,𝑘𝜌+𝛿^𝑑3,𝑘𝐷^𝛼1_3,𝑘1−𝑉₀/𝛿1−𝑉₁/𝛿1−𝑈₀./𝛿 (160)
From (160), (156), and (157), we get that 𝐵_3,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤_{𝛼≥0𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝐷^𝛼1_3,𝑘×𝒟_𝜌,𝛼2Ψ_{𝛼2−𝑆+𝑘}𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2−𝑆+𝑘,𝑉0,𝑉1,(𝑈ℎ)2ℎ∈𝐼(𝛼−𝑆+𝑘)}𝑊^𝛼, (161) where 𝒟_𝜌,𝛼2=𝐷_3,𝑘𝜌+𝛿^𝑑3,𝑘1−𝑉₀/𝛿1−𝑉₁/𝛿1−𝑈₀×()/𝛼/𝛿exp−𝜎𝜌𝑆−𝑘₂+1^𝑏Π^𝑆−𝑘_𝑙=1𝛼₂.−𝑙+1 (162) Using the estimates (125), we deduce that 𝒟_𝜌,𝛼2≤𝐷_3,𝑘𝑑_3,𝑘exp(−1)/𝜎(𝑆−𝑘)^𝑑3,𝑘exp(𝛿𝜎(𝑆−𝑘))1−𝑉₀/𝛿1−𝑉₁/𝛿1−𝑈₀×𝛼/𝛿₂+1^{𝑏𝑑3,𝑘}Π^𝑆−𝑘_𝑙=1𝛼₂.−𝑙+1 (163) Under the assumption (143), we get a constant 𝐶_9.1>0 (depending on 𝐷_3,𝑘,𝑑_3,𝑘,𝑆,𝑘,𝛿,𝜎,𝑉₀,𝑉₁,𝑈₀,𝑏) such that 𝒟_𝜌,𝛼2≤𝐶_9.1 (164) for all 𝜌>1, all 𝛼₂≥𝑆−𝑘. Finally, collecting (161) and (164), we get 𝐵_3,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤∑_𝛼≥0∑_{𝛼1+𝛼2=𝛼,𝛼2≥𝑆−𝑘}𝐷^𝛼1_3,𝑘×𝐶_9.1Ψ_{𝛼2−𝑆+𝑘}𝑉₀,𝑉₁,𝑈_{ℎℎ∈𝐼(𝛼2−𝑆+𝑘)𝜌,𝛼2−𝑆+𝑘,𝑉0,𝑉1,(𝑈ℎ)2ℎ∈𝐼(𝛼−𝑆+𝑘)}𝑊^{𝛼1+𝛼2−𝑆+𝑘}𝑊^𝑆−𝑘=𝐶_9.11−𝐷_3,𝑘𝑊𝑊^𝑆−𝑘Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} (165) which yields (144).

3.2. A Functional Partial Differential Equation in the Banach Spaces of Infinitely Many Variables 𝐺_{(𝜌,𝑉0,𝑉1,(𝑈)ℎ≥0,𝑊)}

In the next proposition, we solve a functional fixed point equation within the Banach spaces of formal series introduced in the previous subsection.

Proposition 19. One makes the following assumptions: 𝑆≥𝑘+1+max𝑏𝑑_1,𝑘+2+3,𝑑+1+𝑏𝑑+𝑑_1,𝑘+1,𝑆≥𝑘+3+𝑏2+𝑑_2,𝑘,𝑆≥𝑘+1+𝑏max𝑑_1,𝑘,𝑑_2,𝑘,𝑆≥𝑘+𝑏𝑑_3,𝑘 (166) for all 𝑘∈𝒮. Then, for given 𝑉₀,𝑉₁,𝛿>0, there exists 𝑊>0 (independent of 𝜌>1) such that, for all 𝐼(̃𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊)∈𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}, the functional equation Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊=_𝑘∈𝒮𝐵_1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝜕_𝑉0Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊+𝐵_1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝔻_𝐀Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊+_𝑘∈𝒮𝐵_2,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝜕_𝑉1Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊+𝐵_2,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝔻_𝐁Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊+_𝑘∈𝒮𝐵_3,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0+𝐼̃𝑉,𝑊₀,𝑉₁,𝑈_ℎℎ≥0,𝑊 (167) has a unique solution Ψ(𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊)∈𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}. Moreover, one has that Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤2𝐼̃𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}. (168)

Proof . We consider the map 𝔐 from the space 𝔾[[𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊]] of formal series (introduced in Definition 4) into itself defined as follows: 𝔐Δ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊=_𝑘∈𝒮𝐵_1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝜕_𝑉0Δ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊+𝐵_1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝔻_𝐀Δ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊+_𝑘∈𝒮𝐵_2,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝜕_𝑉1Δ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊+𝐵_2,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝔻_𝐁Δ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊+_𝑘∈𝒮𝐵_3,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘Δ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊 (169) for all Δ(𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊)∈𝔾[[𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊]].
In order to prove the proposition, we need the following lemma.
Lemma 20. Let id be the identity map 𝑥↦𝑥𝑓𝑟𝑜𝑚  𝔾[[𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊]]   into itself. Then, for a well-chosen 𝑊>0, the map id−𝔐  defines an invertible map such that (id−𝔐)⁻¹  is defined from 𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}  into itself. Moreover, one has that(id−𝔐)⁻¹Ξ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤2Ξ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} (170)for all Ξ(𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊)∈𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}.
Proof. Taking care of the constraints (166), we get from Propositions 15 and 18 a constant 𝐶₁₀>0 (depending on the constants introduced above and also on the aforementioned propositions: 𝑎,max_{0≤𝑝≤𝑑}𝑎_𝑝, 𝛿,𝛿, 𝑏,𝑑, max_𝑘∈𝒮𝑑_1,𝑘, max_𝑘∈𝒮𝐷_1,𝑘, max_𝑘∈𝒮𝑑_2,𝑘,max_𝑘∈𝒮𝐷_2,𝑘, max_𝑘∈𝒮𝑑_3,𝑘,max_𝑘∈𝒮𝐷_3,𝑘, 𝜎,𝜈, 𝑆,𝒮, and 𝜅,𝑉₀,𝑉₁ but independent of 𝜌>1) such that 𝔐Δ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤𝐶_{10𝑘∈𝒮}𝑊^𝑆−𝑘×Δ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} (171) for all Δ(𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊)∈𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} with 0≤𝑊≤min_{𝑚∈{0,1,2},𝑘∈𝒮}1/(2𝐷_𝑚+1,𝑘). Since 𝑆>𝑘 for all 𝑘∈𝒮, we can choose 𝑊>0 such that 𝐶_{10𝑘∈𝒮}𝑊^𝑆−𝑘<12 (172) together with 𝑊≤min_{𝑚∈{0,1,2},𝑘∈𝒮}1/(2𝐷_𝑚+1,𝑘). We deduce that 𝔐Δ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤12Δ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} (173) for all Δ(𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊)∈𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}. This yields the estimates (170).
Finally, let 𝐼(̃𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊)∈𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} for 𝑊>0 chosen as in Lemma 20. We define Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0=(,𝑊id−𝔐)⁻¹𝐼̃𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊. (174)
By construction, Ψ(𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊) belongs to 𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} and solves (167) with the estimates (168).

4. Analytic Solutions with Growth Estimates of Linear Partial Differential Equations in ℂ³

We are now in position to state the main result of our work.

Theorem 21. Let 𝑏_𝑚,𝑘(𝑡,𝑧,𝑢₀,𝑤) be the functions defined in (15) for 𝑚=1,2,3 and 𝑘∈𝒮. Let one assume that there exists 𝑏>1 such that 𝑆≥𝑘+1+max𝑏𝑑_1,𝑘+2+3,𝑑+1+𝑏𝑑+𝑑_1,𝑘+1,𝑆≥𝑘+3+𝑏2+𝑑_2,𝑘,𝑆≥𝑘+1+𝑏max𝑑_1,𝑘,𝑑_2,𝑘,𝑆≥𝑘+𝑏𝑑_3,𝑘 (175) for all 𝑘∈𝒮. For all 0≤𝑗≤𝑆−1, one considers functions 𝜔_𝑗(𝑡,𝑧) which are assumed to be holomorphic and bounded on the product 𝐷(0,𝑅^′)².
Then, there exist constants 𝜎,𝑊,𝐶₁₂>0 such that the problem 𝜕^𝑆_𝑤𝑌(𝑡,𝑧,𝑤)=_𝑘∈𝒮𝑏_1,𝑘(𝑡,𝑧,𝑋(𝑡,𝑧),𝑤)𝜕_𝑡𝜕^𝑘_𝑤𝑌(𝑡,𝑧,𝑤)+𝑏_2,𝑘(𝑡,𝑧,𝑋(𝑡,𝑧),𝑤)𝜕_𝑧𝜕^𝑘_𝑤𝑌(𝑡,𝑧,𝑤)+𝑏_3,𝑘(𝑡,𝑧,𝑋(𝑡,𝑧),𝑤)𝜕^𝑘_𝑤𝑌(𝑡,𝑧,𝑤) (176) with initial data 𝜕^𝑗_𝑤𝑌(𝑡,𝑧,0)=𝜔_𝑗(𝑡,𝑧),0≤𝑗≤𝑆−1, (177) has a solution 𝑌(𝑡,𝑧,𝑤) which is holomorphic on Int(𝐾)×𝐷(0,𝑊/2) and which fulfills the following estimates: sup_{()∈𝑡,𝑧Int(𝐾),𝑤∈𝐷(0,𝑊/2)}|𝑌(𝑡,𝑧,𝑤)|≤𝐶₁₂exp𝜎𝜁(𝑏)𝜌+^𝑆−1_𝑗=0sup_{()∈𝑡,𝑧Int(𝐾)}𝜔_𝑗(𝑡,𝑧)𝑊/2^𝑗,𝑗! (178) where 𝜁(𝑏)=∑_𝑛≥01/(𝑛+1)^𝑏, for any compact set 𝐾⊂𝐷(0,𝑅)²⧵Θ with nonempty interior Int(𝐾) for some 𝑅<𝑅^′ and any 𝜌>1 which satisfies (10). One stresses that the constants 𝜎,𝑊,𝐶₁₂>0 do not depend neither on 𝐾 nor on 𝜌>1.

Proof. By convention, we will put 𝜔_𝑗(𝑡,𝑧)≡0 for all 𝑗≥𝑆. On the other hand, we specialize the functions 𝜔_𝛼 which were introduced in (12) in order that 𝜔_𝛼𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼)}=𝜔_𝛼𝑣₀,𝑣₁,𝑢₀=_{𝑘∈𝒮𝛼1+𝛼2=𝛼}𝑏𝛼!^1,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀𝛼₁!𝜕_𝑣0𝜔_𝛼2+𝑘𝑣₀,𝑣₁𝛼₂!+𝑏_2,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀𝛼₁!𝜕_𝑣1𝜔_𝛼2+𝑘𝑣₀,𝑣₁𝛼₂!+𝑏_3,𝑘,𝛼1𝑣₀,𝑣₁,𝑢₀𝛼₁!𝜔_𝛼2+𝑘𝑣₀,𝑣₁𝛼₂!. (179) By construction and using the definition (26), we can write with the help of the Kronecker symbol, 𝜔_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}=𝜔_{𝛼,𝑛0,𝑛1,𝑙0}×Π_{ℎ∈𝐼(𝛼)⧵{0}}𝛿_0,𝑙ℎ, (180) where 𝜔_{𝛼,𝑛0,𝑛1,𝑙0}=sup_{|𝑣0|<𝑅,|𝑣1|<𝑅,|𝑢0|<𝜌}𝜕^𝑛0_𝑣0𝜕^𝑛1_𝑣1𝜕^𝑙0_𝑢0𝜔_𝛼𝑣₀,𝑣₁,𝑢₀. (181)
Lemma 22. There exist 𝑉₀,𝑉₁,𝑊>0  such that the formal seriesΩ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊=_{𝛼≥0𝑛0,𝑛1,𝑙ℎ(𝛼)≥0,ℎ∈𝐼}𝜔_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}×𝑉^𝑛0₀𝑛₀!𝑉^𝑛1₁𝑛₁!Π_{ℎ∈𝐼(𝛼)}𝑈^𝑙ℎ_ℎ𝑙_ℎ!𝑊^𝛼𝛼! (182)belongs to 𝐺_(𝜌,𝑉_0,𝑉_{1,(𝑈ℎ)ℎ≥0,}𝑊₎. Moreover, there exists a constant 𝐶₁₁>0  (independent of 𝜌) such thatΩ(𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊)_(𝜌,𝑉_0,𝑉_{1,(𝑈ℎ)ℎ≥0,}𝑊₎≤𝐶₁₁. (183)
Proof. Let 𝑘∈𝒮. Due to the estimates (14) for the functions 𝑏_{𝑚,𝑘,𝛼}(𝑡,𝑧,𝑢₀), we get couples of constants 𝐷_1,𝑘,𝐷_1,𝑘>0, 𝐷_2,𝑘,𝐷_2,𝑘>0, and 𝐷_3,𝑘,𝐷_3,𝑘>0 such that 𝑏_1,𝑘,𝛼1𝜒₀,𝜒₁,𝜉₀≤𝐷_1,𝑘𝜌+𝛿^𝑑1,𝑘𝛼₁!𝐷_1,𝑘^𝛼1,𝑏_2,𝑘,𝛼1𝜒₀,𝜒₁,𝜉₀≤𝐷_2,𝑘𝜌+𝛿^𝑑2,𝑘𝛼₁!𝐷_2,𝑘^𝛼1,𝑏_3,𝑘,𝛼1𝜒₀,𝜒₁,𝜉₀≤𝐷_3,𝑘𝜌+𝛿^𝑑3,𝑘𝛼₁!𝐷_3,𝑘^𝛼1 (184) for all 𝛼₁≥0, all |𝜒₀|<𝑅+𝛿<𝑅^′, |𝜒₁|<𝑅+𝛿<𝑅^′, |𝜉₀|<𝜌+𝛿. Moreover, we also get couples of constants 𝐸_1,𝑘,𝐸_1,𝑘>0, 𝐸_2,𝑘,𝐸_2,𝑘>0, and 𝐸_3,𝑘,𝐸_3,𝑘>0 such that 𝜕_𝜒0𝜔_𝛼2+𝑘𝜒₀,𝜒₁≤𝐸_1,𝑘𝛼₂!(𝐸_1,𝑘)^𝛼2,𝜕_𝜒1𝜔_𝛼2+𝑘𝜒₀,𝜒₁≤𝐸_2,𝑘𝛼₂!(𝐸_2,𝑘)^𝛼2,𝜔_𝛼2+𝑘𝜒₀,𝜒₁≤𝐸_3,𝑘𝛼₂!𝐸_3,𝑘^𝛼2 (185) for all 𝛼₂≥0, all |𝜒₀|<𝑅+𝛿<𝑅^′, |𝜒₁|<𝑅+𝛿<𝑅^′. From (184) and (185) we deduce 𝜔_𝛼𝜒₀,𝜒₁,𝜉₀≤_{𝑘∈𝒮𝛼1+𝛼2=𝛼}𝛼!𝐷_1,𝑘𝐸_1,𝑘𝜌+𝛿^𝑑1,𝑘𝐷_1,𝑘^𝛼1𝐸_1,𝑘^𝛼2+𝐷_2,𝑘𝐸_2,𝑘𝜌+𝛿^𝑑2,𝑘𝐷_2,𝑘^𝛼1𝐸_2,𝑘^𝛼2+𝐷_3,𝑘𝐸_3,𝑘𝜌+𝛿^𝑑3,𝑘𝐷_3,𝑘^𝛼1𝐸_3,𝑘^𝛼2 (186) for all 𝛼≥0, all |𝜒₀|<𝑅+𝛿<𝑅^′, |𝜒₁|<𝑅+𝛿<𝑅^′, |𝜉₀|<𝜌+𝛿. From the Cauchy formula in several variables, one can write 𝜕^𝑛0_𝑣0𝜕^𝑛1_𝑣1𝜕^𝑙0_𝑢0𝜔_𝛼𝑣₀,𝑣₁,𝑢₀𝑛₀!𝑛₁!𝑙₀!=12𝑖𝜋³×_{𝐶(𝑣0,𝛿)𝐶(𝑣1,𝛿)𝐶(𝑢0,𝛿)}𝜔_𝛼𝜒₀,𝜒₁,𝜉₀×𝑑𝜒₀𝑑𝜒₁𝑑𝜉₀×𝜒₀−𝑣₀^𝑛0+1×𝜒₁−𝑣₁^𝑛1+1×𝜉₀−𝑢₀^𝑙0+1−1 (187) for all |𝑣₀|<𝑅, |𝑣₁|<𝑅, |𝑢₀|<𝜌. We deduce that 𝜔_{𝛼,𝑛0,𝑛1,𝑙0}𝑛₀!𝑛₁!𝑙₀!≤1𝛿^{𝑛0+𝑛1+𝑙0}×_{𝑘∈𝒮𝛼1+𝛼2=𝛼}×𝐷𝛼!_1,𝑘𝐸_1,𝑘𝜌+𝛿^𝑑1,𝑘𝐷_1,𝑘^𝛼1𝐸_1,𝑘^𝛼2+𝐷_2,𝑘𝐸_2,𝑘𝜌+𝛿^𝑑2,𝑘𝐷_2,𝑘^𝛼1𝐸_2,𝑘^𝛼2+𝐷_3,𝑘𝐸_3,𝑘𝜌+𝛿^𝑑3,𝑘𝐷_3,𝑘^𝛼1𝐸_3,𝑘^𝛼2 (188) for all 𝛼≥0, all 𝑛₀,𝑛₁,𝑙₀≥0. Using (180), we get that Ω(𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊)_(𝜌,𝑉_0,𝑉_{1,(𝑈ℎ)ℎ≥0,}𝑊₎=_{𝛼≥0𝑛0,𝑛1,𝑙0≥0}𝑤_{𝛼,𝑛0,𝑛1,𝑙0}exp𝜎𝑟_𝑏(𝛼)𝜌×𝑉^𝑛0₀𝑉^𝑛1₁𝑈^𝑙0₀𝑛₀+𝑛₁+𝑙₀!𝑊+𝛼^𝛼. (189) From (188), (125), and with the help of the classical estimates 𝑛₀+𝑛₁+𝑙₀+𝛼!≥𝑛₀!𝑛₁!𝑙₀!𝛼!, (190) for all 𝑛₀,𝑛₁,𝑙₀,𝛼≥0, we get a constant 𝐶_11,1>0 (depending on 𝐷_1,𝑘, 𝑑_1,𝑘, 𝐸_1,𝑘, 𝐷_2,𝑘, 𝑑_2,𝑘, 𝐸_2,𝑘, 𝐷_3,𝑘, 𝑑_3,𝑘, 𝐸_3,𝑘 for all 𝑘∈𝒮, 𝜎, 𝛿)  such that Ω𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊_(𝜌,𝑉₀^,𝑉₁^,(𝑈_ℎ⁾_ℎ≥0^,𝑊⁾≤𝐶_11,11−𝑉₀/𝛿1−𝑉₁/𝛿1−𝑈₀/𝛿×_𝑘∈𝒮11−𝐷_1,𝑘𝑊1−𝐸_1,𝑘𝑊+11−𝐷_2,𝑘𝑊1−𝐸_2,𝑘𝑊+11−𝐷_3,𝑘𝑊1−𝐸_3,𝑘𝑊. (191) We choose 0<𝑊<min_𝑘∈𝒮12𝐷_1,𝑘,12𝐷_2,𝑘,12𝐷_3,𝑘,12𝐸_1,𝑘,12𝐸_2,𝑘,12𝐸_3,𝑘,0<𝑉₀<𝛿2,0<𝑉₁<𝛿2,0<𝑈₀<𝛿2. (192) From (191) we deduce the inequality (183).
Under the assumption (175), we get from Proposition 19 four constants 0<𝑉₀<𝑉₀, 0<𝑉₁<𝑉₁, 0<𝑈₀, and 0<𝑊<𝑊 (independent of 𝜌) such that the functional equation Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊=_𝑘∈𝒮𝐵_1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝜕_𝑉0Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊+𝐵_1,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝔻_𝐀Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊+_𝑘∈𝒮𝐵_2,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝜕_𝑉1Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊+𝐵_2,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘𝔻_𝐁Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0,𝑊+_𝑘∈𝒮𝐵_3,𝑘𝑉₀,𝑉₁,𝑈₀,𝑊𝜕_𝑊^−𝑆+𝑘Ψ𝑉₀,𝑉₁,𝑈_ℎℎ≥0+Ω𝑉,𝑊₀,𝑉₁,𝑈_ℎℎ≥0,𝑊 (193) has a unique solution Ψ(𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊) belonging to 𝐺_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)} which satisfies moreover the estimates Ψ(𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊)_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤2Ω(𝑉₀,𝑉₁,(𝑈_ℎ)_ℎ≥0,𝑊)_{(𝜌,𝑉0,𝑉1,(𝑈ℎ)ℎ≥0,𝑊)}≤2𝐶₁₁. (194) Now, from Proposition 6, we know that the sequence 𝜑_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)} introduced in (25) satisfies the inequality 𝜑_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)}≤𝜓_{𝛼,𝑛0,𝑛1,(𝑙ℎ)ℎ∈𝐼(𝛼)} (195) for all 𝛼≥0, all 𝑛₀,𝑛₁,𝑙_ℎ≥0, for ℎ∈𝐼(𝛼). Gathering (194) and (195) and from the definition of the Banach spaces in Section 3.1, we get, in particular, for 𝑛₀=𝑛₁=𝑙_ℎ=0, for all ℎ∈𝐼(𝛼), all 𝛼≥0, that sup_{|𝑣0|<𝑅,|𝑣1|<𝑅,|𝑢ℎ|<𝜌,ℎ∈𝐼(𝛼)}𝜙_𝛼𝑣₀,𝑣₁,𝑢_{ℎℎ∈𝐼(𝛼)}≤𝜓_{𝛼,0,0,(0)ℎ∈𝐼(𝛼)}≤2𝐶₁₁exp𝜎𝑟_𝑏(𝛼)𝜌1𝑊^𝛼𝛼!≤2𝐶₁₁(𝑏)𝜌1exp𝜎𝜁𝑊^𝛼𝛼! (196) for all 𝛼≥0 and where 𝜁(𝑏)=∑_𝑛≥01/(𝑛+1)^𝑏. From (196), we get that the formal series 𝑈(𝑡,𝑧,𝑤) introduced in (11) actually defines a holomorphic function (denoted again by 𝑈(𝑡,𝑧,𝑤)) on Int(𝐾)×𝐷(0,𝑊/2) for which the estimates sup_{(𝑡,𝑧)∈Int(𝐾),𝑤∈𝐷(0,𝑊/2)}|𝑈(𝑡,𝑧,𝑤)|≤4𝐶₁₁exp𝜎𝜁(𝑏)𝜌 (197) hold and which satisfies (17) on Int(𝐾)×𝐷(0,𝑊/2).
Finally, we define the function 𝑌(𝑡,𝑧,𝑤)=𝜕_𝑤^−𝑆𝑈(𝑡,𝑧,𝑤)+^𝑆−1_𝑗=0𝜔_𝑗()𝑤𝑡,𝑧^𝑗.𝑗! (198) By construction, 𝑌(𝑡,𝑧,𝑤) defines a holomorphic function on Int(𝐾)×𝐷(0,𝑊/2) with bounds estimates sup_{(𝑡,𝑧)∈Int(𝐾),𝑤∈𝐷(0,𝑊/2)}|𝑌(𝑡,𝑧,𝑤)|≤4𝑊2^𝑆𝐶₁₁(𝑏)𝜌+exp𝜎𝜁_{𝑆−1𝑗=0}sup_{(𝑡,𝑧)∈Int(𝐾)}𝜔_𝑗(𝑡,𝑧)𝑊/2^𝑗𝑗! (199) and solves the problem (176), (177). This yields the result.

Acknowledgments

A. Lastra is partially supported by Project MTM2012-31439 of Ministerio de Ciencia e Innovacion, Spain. S. Malek is partially supported by the French ANR-10-JCJC 0105 project and the PHC Polonium 2013 Project no. 28217SG.