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On complex singularity analysis for some linear partial differential equations in C3

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(1)Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 394564, 30 pages http://dx.doi.org/10.1155/2013/394564. Research Article On Complex Singularity Analysis for Some Linear Partial Differential Equations in C3 A. Lastra,1 S. Malek,2 and C. Stenger3 1. Facultad de Ciencias, University of Alcalá, Apartado de Correos 20, 28871 Alcalá de Henares (Madrid), Spain UFR de Mathématiques, University of Lille 1, Cité Scientifique M2, 59655 Villeneuve d’Ascq Cedex, France 3 Laboratoire Mathématiques, Images et Applications, University of La Rochelle, Avenue Michel Crépeau, 17042 La Rochelle Cedex, France 2. Correspondence should be addressed to A. Lastra; alberto.lastra@uah.es Received 1 April 2013; Accepted 4 June 2013 Academic Editor: Graziano Crasta Copyright © 2013 A. Lastra et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 𝜕𝑤𝑆 𝑌 (𝑡, 𝑧, 𝑤) = ∑ (𝑎1,𝑘 (𝑡, 𝑧, 𝑤) 𝜕𝑡 𝜕𝑤𝑘 𝑌 (𝑡, 𝑧, 𝑤) 𝑘∈S. + 𝑎2,𝑘 (𝑡, 𝑧, 𝑤) 𝜕𝑧 𝜕𝑤𝑘 𝑌 (𝑡, 𝑧, 𝑤). (1). +𝑎3,𝑘 (𝑡, 𝑧, 𝑤) 𝜕𝑤𝑘 𝑌 (𝑡, 𝑧, 𝑤)) for given initial data 𝜕𝑤𝑗 𝑌(𝑡, 𝑧, 0) = 𝜑𝑗 (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 𝑎𝑚,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 𝜑𝑗 (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 𝑎𝑚,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.

(2) 2. Abstract and Applied Analysis. 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 𝑙. 𝑢 (𝑡, 𝑧) = ∑𝑢𝑙 (𝑡, 𝑧) (𝑢 (𝑡)) , 𝑙≥0. (2). where 𝑢𝑙 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 𝐶 exp(𝑀|𝑡 − 𝑡0 |−𝜇 ) for some constants 𝐶, 𝑀, 𝜇 > 0. In this work, the coefficients 𝑎𝑚,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 𝑎𝑚,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) = ∑ 𝜙𝛼 (𝑡, 𝑧, ( 𝛼≥0. 𝜕𝑧ℎ 𝑋 (𝑡, 𝑧) 𝑤𝛼 ) ) , 𝛼! ℎ!]ℎ 0≤ℎ≤𝛼. (3). solutions of some auxiliary nonhomogeneous integrodifferential equation (17) with polynomial coefficients in 𝑋(𝑡, 𝑧). The coefficients 𝜙𝛼 , 𝛼 ≥ 0, are holomorphic functions on some polydisc in C𝛼+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 𝜙𝛼 denoted by 𝜑𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )0≤ℎ≤𝛼 for all nonnegative integers 𝛼, 𝑛0 , 𝑛1 , 𝑙ℎ with 0 ≤ ℎ ≤ 𝛼 (Proposition 3). In the next section, we construct a sequence of coefficients 𝜓𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )0≤ℎ≤𝛼 which is larger than the latter sequence 𝜑𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )0≤ℎ≤𝛼 ≤ 𝜓𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )0≤ℎ≤𝛼. (4). for any nonnegative integers 𝛼, 𝑛0 , 𝑛1 , 𝑙ℎ 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−𝑆 𝑈(𝑡, 𝑧, 𝑤) and a polynomial in 𝑤 having the initial data 𝜑𝑗 as coefficients. Owing to the fact that the generating series of the sequence 𝜓𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )0≤ℎ≤𝛼 , solution of (51), belongs to the Banach spaces mentioned above, we get estimates for the holomorphic functions 𝜙𝛼 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.

(3) Abstract and Applied Analysis. 3. 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).. 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+ 𝑒𝑖𝜃 with 𝜃 ∈ R depending on the choice of the determination of the logarithm.. 2. Formal Series Solutions of Linear Integrodifferential Equations. 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. 2.1. Some Nonlinear Partial Differential Equation. We consider the following nonlinear partial differential equation:. sup |𝑋 (𝑡, 𝑧)| ≤. 𝑑. 𝜕𝑡 𝑋 (𝑡, 𝑧) = 𝑎 (𝑡, 𝑧) 𝜕𝑧 𝑋 (𝑡, 𝑧) + ∑ 𝑎𝑝 (𝑡, 𝑧) 𝑋𝑝 (𝑡, 𝑧) ,. (5). 𝑝=0. where 𝑑 ≥ 2 is some integer and the coefficients 𝑎(𝑡, 𝑧), 𝑎𝑝 (𝑡, 𝑧) are holomorphic functions on some polydisc 𝐷(0, 𝑅󸀠 )2 ⊂ C2 such that 𝑎𝑑 (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 𝜕𝑡 𝑢 (𝑡, 𝑥) = 𝐹 (𝑡, 𝑥, 𝑢 (𝑡, 𝑥) , 𝜕𝑥 𝑢 (𝑡, 𝑥)). (6). 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 1 (𝜕 𝑔 (𝑡, 𝑧) − 𝜕𝑡 𝑔 (𝑡, 𝑧)) 𝑛 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 𝜕𝑡 𝑋 (𝑡, 𝑧) = 𝜕𝑧 𝑋 (𝑡, 𝑧) + 𝑎𝑛+1 (𝑡, 𝑧) 𝑋. 𝑛+1. (𝑡, 𝑧). (8). (9). Let 𝐾 ⊊ 𝐾0 be a compact set with nonempty interior Int(𝐾). From the Cauchy formula, there exists a real number ] > 0 such that 󵄨󵄨 ℎ 󵄨 󵄨󵄨𝜕𝑧 𝑋 (𝑡, 𝑧)󵄨󵄨󵄨 𝜌 󵄨≤ (10) sup 󵄨 ℎ 2 ℎ!] (𝑡,𝑧)∈Int(𝐾) for all integers ℎ ≥ 0. For all integers 𝛼 ≥ 0, we denote 𝐼(𝛼) = {0, . . . , 𝛼}. We consider a sequence of functions 𝜙𝛼 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼) ) which are holomorphic and bounded on the polydisc 𝐷(0, 𝑅)2 Πℎ∈𝐼(𝛼) 𝐷(0, 𝜌), for all 𝛼 ≥ 0. We define the formal series in the 𝑤 variable as 𝑈 (𝑡, 𝑧, 𝑤) = ∑ 𝜙𝛼 (𝑡, 𝑧, ( 𝛼≥0. for (𝑡, 𝑥) ∈ C × C𝑛 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).. 𝑎𝑛+1 (𝑡, 𝑧) =. (𝑡,𝑧)∈𝐾0. 𝜌 . 2. 𝜕𝑧ℎ 𝑋 (𝑡, 𝑧) 𝑤𝛼 ) ) . 𝛼! ℎ!]ℎ ℎ∈𝐼(𝛼). (11). For all 𝛼 ≥ 0, we consider a holomorphic and bound̃𝛼 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼) ) on the product 𝐷(0, 𝑅󸀠 )2 ed function 𝜔 Πℎ∈𝐼(𝛼) 𝐷(0, 𝜌). We define the formal series ̃ (𝑡, 𝑧, 𝑤) = ∑ 𝜔 ̃𝛼 (𝑡, 𝑧, ( 𝜔 𝛼≥0. 𝜕𝑧ℎ 𝑋 (𝑡, 𝑧) 𝑤𝛼 ) ) . 𝛼! ℎ!]ℎ ℎ∈𝐼(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 𝑏𝑚,𝑘,𝛼 (𝑡, 𝑧, 𝑢0 ) which is holomorphic on 𝐷(0, 𝑅󸀠 )2 × C and satisfies estimates of the following form. ̂𝑚,𝑘 > 0 and an integer There exist two constants 𝐷𝑚,𝑘 > 0, 𝐷 𝑑𝑚,𝑘 ≥ 0 such that sup. |𝑡|<𝑅󸀠 ,|𝑧|<𝑅󸀠 ,|𝑢0 |≤𝜌. 󵄨󵄨 󵄨 𝑑 ̂𝛼 󵄨󵄨𝑏𝑚,𝑘,𝛼 (𝑡, 𝑧, 𝑢0 )󵄨󵄨󵄨 ≤ 𝐷𝑚,𝑘 𝜌 𝑚,𝑘 𝐷 𝑚,𝑘 𝛼!. (14). for all 𝛼 ≥ 0, with all 𝜌 ≥ 1. In particular, each function 𝑢0 󳨃→ 𝑏𝑚,𝑘,𝛼 (𝑡, 𝑧, 𝑢0 ) is a polynomial of degree at most 𝑑𝑚,𝑘 for all (𝑡, 𝑧) ∈ 𝐷(0, 𝑅󸀠 )2 . Finally, for all 𝑘 ∈ S, 𝑚 = 1, 2, 3, we consider the series 𝑏𝑚,𝑘 (𝑡, 𝑧, 𝑢0 , 𝑤) = ∑ 𝑏𝑚,𝑘,𝛼 (𝑡, 𝑧, 𝑢0 ) 𝛼≥0. 𝑤𝛼 𝛼!. (15). which define holomorphic functions on 𝐷(0, 𝑅󸀠 )2 × C × ̂𝑚,𝑘 . 𝐷(0, 𝑤), for any 0 < 𝑤 ≤ 1/𝐷.

(4) 4. Abstract and Applied Analysis +𝑏3,𝑘 (𝑡, 𝑧, 𝑋 (𝑡, 𝑧) , 𝑤) 𝜕𝑤−𝑆+𝑘 𝑈 (𝑡, 𝑧, 𝑤)). Proposition 2. Assume that the sequence of functions (𝜙𝛼 )𝛼≥0 satisfies the following recursion:. ̃ (𝑡, 𝑧, 𝑤) +𝜔. 𝜙𝛼 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼) ). (17). 𝛼! = ∑. ∑. 𝛼1 !. 𝑘∈S 𝛼1 +𝛼2 =𝛼,𝛼2 ≥𝑆−𝑘. ×(. for all (𝑡, 𝑧) ∈ Int(𝐾), where 𝜕𝑤−𝑚 denotes the 𝑚-iterate of the 𝑤 usual integration operator ∫0 [⋅]𝑑𝑠.. 𝑏1,𝑘,𝛼1 (V0 , V1 , 𝑢0 ). Proof. We have that. 𝜕V0 𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) ). 𝑏3,𝑘 (𝑡, 𝑧, 𝑋 (𝑡, 𝑧) , 𝑤) 𝜕𝑤−𝑆+𝑘 𝑈 (𝑡, 𝑧, 𝑤). 𝛼2 ! +. ∑. 𝑗∈𝐼(𝛼2 +𝑘−𝑆). 𝜕V𝑙11 𝑎 (V0 , V1 ). ( ∑. 𝑙1 !]𝑙1. 𝑙1 +𝑙2 =𝑗 𝑑. +∑. ∑. (𝑙2 + 1) ]𝑢𝑙2 +1. = ∑(. ∑. 𝛼≥0. 𝛼1 +𝛼2 =𝛼,𝛼2 ≥𝑆−𝑘. 𝜕V𝑗10 𝑎𝑝 (V0 , V1 ). 𝑝=0𝑗0 +⋅⋅⋅+𝑗𝑝 =𝑗. 𝑗0. ×. !]𝑗0. 𝑝. × Π𝑙=1 𝑢𝑗𝑙 ). ×. + ∑. 𝜕𝑢𝑗 𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) ) 𝛼2 !. ∑. ∑ 𝑗∈𝐼(𝛼2 +𝑘−𝑆). 𝛼2 !. 𝑘∈S 𝛼1 +𝛼2 =𝛼,𝛼2 ≥𝑆−𝑘. × +. = ∑( 𝛼≥0. 𝑤𝛼 , 𝛼!. ). (18). 𝛼!. ∑. 𝑏2,𝑘,𝛼1 (𝑡, 𝑧, 𝑋 (𝑡, 𝑧)) 𝛼1 !. 𝛼1 +𝛼2 =𝛼,𝛼2 ≥𝑆−𝑘. ×. 𝜕𝑧 (𝜙𝛼2 +𝑘−𝑆 (𝑡, 𝑧, (𝜕𝑧ℎ 𝑋 (𝑡, 𝑧) /ℎ!]ℎ )ℎ∈𝐼(𝛼 +𝑘−𝑆) )) 2. 𝛼2 !. ). 𝑏3,𝑘,𝛼1 (V0 , V1 , 𝑢0 ). ∑. 2. 𝛼2 !. and we also see that. (𝑗 + 1) ]𝑢𝑗+1. 𝜕𝑢𝑗 𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) ). 𝛼1 !. 𝜙𝛼2 +𝑘−𝑆 (𝑡, 𝑧, (𝜕𝑧ℎ 𝑋 (𝑡, 𝑧) /ℎ!]ℎ )ℎ∈𝐼(𝛼 +𝑘−𝑆) ). 𝛼1 !. 𝛼2 !. ×. 𝑏3,𝑘,𝛼1 (𝑡, 𝑧, 𝑋 (𝑡, 𝑧)). 𝑏2,𝑘 (𝑡, 𝑧, 𝑋 (𝑡, 𝑧) , 𝑤) 𝜕𝑧 𝜕𝑤−𝑆+𝑘 𝑈 (𝑡, 𝑧, 𝑤). 𝜕V1 𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) ) +. +∑. ). 𝑏2,𝑘,𝛼1 (V0 , V1 , 𝑢0 ). 𝑘∈S 𝛼1 +𝛼2 =𝛼,𝛼2 ≥𝑆−𝑘. ×(. ×. 𝛼!. ×. 𝛼1 !. 𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) ). (19). with. 𝛼2 !. 𝜕𝑧 (𝜙𝛼2 +𝑘−𝑆 (𝑡, 𝑧, (. ̃𝛼 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼) ) 𝜔 𝛼!. 𝑤𝛼 𝛼!. ). (16). for all 𝛼 ≥ 0, all V0 , V1 ∈ 𝐷(0, 𝑅), all 𝑢ℎ ∈ 𝐷(0, 𝜌), for ℎ ∈ 𝐼(𝛼). Then, the formal series 𝑈(𝑡, 𝑧, 𝑤) satisfies the following integrodifferential equation: 𝑈 (𝑡, 𝑧, 𝑤) = ∑ (𝑏1,𝑘 (𝑡, 𝑧, 𝑋 (𝑡, 𝑧) , 𝑤) 𝜕𝑡 𝜕𝑤−𝑆+𝑘 𝑈 (𝑡, 𝑧, 𝑤) 𝑘∈S. + 𝑏2,𝑘 (𝑡, 𝑧, 𝑋 (𝑡, 𝑧) , 𝑤) 𝜕𝑧 𝜕𝑤−𝑆+𝑘 𝑈 (𝑡, 𝑧, 𝑤). 𝜕𝑧ℎ 𝑋 (𝑡, 𝑧) ) )) ℎ!]ℎ ℎ∈𝐼(𝛼2 +𝑘−𝑆). = (𝜕V1 𝜙𝛼2 +𝑘−𝑆 ) (𝑡, 𝑧, ( +. ∑. 𝑗∈𝐼(𝛼2 +𝑘−𝑆). 𝜕𝑧ℎ 𝑋 (𝑡, 𝑧) ) ) ℎ!]ℎ ℎ∈𝐼(𝛼2 +𝑘−𝑆). 𝜕𝑗+1 𝑋 (𝑡, 𝑧) (𝜕 𝜙 ) (𝑗 + 1) ] 𝑧 (𝑗 + 1)!]𝑗+1 𝑢𝑗 𝛼2 +𝑘−𝑆. × (𝑡, 𝑧, (. 𝜕𝑧ℎ 𝑋 (𝑡, 𝑧) ) ), ℎ!]ℎ ℎ∈𝐼(𝛼2 +𝑘−𝑆). (20).

(5) Abstract and Applied Analysis. 5. for all (𝑡, 𝑧) ∈ Int(𝐾). We also get that. for all (𝑡, 𝑧) ∈ Int(𝐾). Finally, gathering all the equalities above and using the recursion (16), one gets the integrodifferential equation (17).. 𝑏1,𝑘 (𝑡, 𝑧, 𝑋 (𝑡, 𝑧) , 𝑤) 𝜕𝑡 𝜕𝑤−𝑆+𝑘 𝑈 (𝑡, 𝑧, 𝑤) = ∑(. ∑. 𝛼≥0. 𝛼1 +𝛼2 =𝛼,𝛼2 ≥𝑆−𝑘. ×. ×. 𝛼!. 𝑏1,𝑘,𝛼1 (𝑡, 𝑧, 𝑋 (𝑡, 𝑧)) 𝛼1 !. 𝜕𝑡 (𝜙𝛼2 +𝑘−𝑆 (𝑡, 𝑧, (𝜕𝑧ℎ 𝑋 (𝑡, 𝑧) /ℎ!]ℎ )ℎ∈𝐼(𝛼 +𝑘−𝑆) )) 2. 𝛼2 !. 𝑤𝛼 𝛼!. ). (21). with 𝜕ℎ 𝑋 (𝑡, 𝑧) 𝜕𝑡 (𝜙𝛼2 +𝑘−𝑆 (𝑡, 𝑧, ( 𝑧 ℎ ) ℎ!] =. +. ∑. 𝑗∈𝐼(𝛼2 +𝑘−𝑆). × (𝑡, 𝑧, (. 𝜑𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼) =. )). sup. |V0 |<𝑅,|V1 |<𝑅,|𝑢ℎ. 󵄨󵄨 𝑛0 𝑛1 󵄨󵄨𝜕V 𝜕V Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ ℎ 󵄨 0 1 |<𝜌,ℎ∈𝐼(𝛼) 󵄨 × 𝜙𝛼 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼) )󵄨󵄨󵄨󵄨. ℎ∈𝐼(𝛼2 +𝑘−𝑆). 𝜕ℎ 𝑋 (𝑡, 𝑧) (𝜕V0 𝜙𝛼2 +𝑘−𝑆 ) (𝑡, 𝑧, ( 𝑧 ℎ ) ℎ!] 𝜕𝑡 𝜕𝑧𝑗 𝑋 (𝑡, 𝑧) 𝑗!]𝑗. 2.3. Recursion for the Derivatives of the Functions 𝜙𝛼 , 𝛼 ≥ 0. We consider a sequence of functions 𝜙𝛼 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼) ), 𝛼 ≥ 0, which are holomorphic and bounded on some polydisc 𝐷(0, 𝑅)2 Πℎ∈𝐼(𝛼) 𝐷(0, 𝜌) for some real numbers 𝑅 > 0 and 𝜌 > 1 and which satisfy the equalities (16). We introduce the sequence. ) ℎ∈𝐼(𝛼2 +𝑘−𝑆). (25). (22). (𝜕𝑢𝑗 𝜙𝛼2 +𝑘−𝑆 ). for all 𝑛0 , 𝑛1 ≥ 0, all 𝑙ℎ ≥ 0, ℎ ∈ 𝐼(𝛼), for all 𝛼 ≥ 0. We define also the following sequences:. 𝜕𝑧ℎ 𝑋 (𝑡, 𝑧) ) ), ℎ!]ℎ ℎ∈𝐼(𝛼2 +𝑘−𝑆). 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 𝜕𝑧𝑗 𝑋 (𝑡, 𝑧) 𝑗!]𝑗. 𝑏𝑚,𝑘,𝛼,𝑛0 ,𝑛1 ,𝑙0 =. 󵄨󵄨 𝑛0 𝑛1 𝑙0 󵄨 󵄨󵄨𝜕V 𝜕V 𝜕𝑢 𝑏𝑚,𝑘,𝛼 (V0 , V1 , 𝑢0 )󵄨󵄨󵄨 , 0 1 0 󵄨 󵄨 |V |<𝑅,|V |<𝑅,|𝑢 |<𝜌 sup. 0. 1. 0. ̃𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼) 𝜔. 𝜕𝑗 𝜕 𝑋 (𝑡, 𝑧) = 𝑧 𝑡 𝑗 𝑗!]. =. sup. |V0 |<𝑅,|V1 |<𝑅,|𝑢ℎ. 󵄨󵄨 𝑛0 𝑛1 󵄨󵄨𝜕V 𝜕V Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ ℎ 󵄨 0 1 |<𝜌,ℎ∈𝐼(𝛼) 󵄨 ̃𝛼 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼) )󵄨󵄨󵄨󵄨 ×𝜔. 𝑑 1 𝑗 = 𝜕 (𝑎 𝑧) 𝜕 𝑋 𝑧) + 𝑎𝑝 (𝑡, 𝑧) 𝑋𝑝 (𝑡, 𝑧)) , ∑ (𝑡, (𝑡, 𝑧 𝑗!]𝑗 𝑧 𝑝=0. (26). (23) and from the Leibniz formula, we can write. for 𝑚 = 1, 2, 3 and 𝑘 ∈ S. We put. 1 𝑗 𝜕 (𝑎 (𝑡, 𝑧) 𝜕𝑧 𝑋 (𝑡, 𝑧)) 𝑗!]𝑗 𝑧 𝜕𝑧𝑙1 𝑎 (𝑡, 𝑧) 𝜕𝑧𝑙2 +1 𝑋 (𝑡, 𝑧) (𝑙 + 1) ] , 2 𝑙1 (𝑙2 + 1)!]𝑙2 +1 𝑙 +𝑙 =𝑗 𝑙1 !]. 𝐴 𝑗 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼+1) ). = ∑ 1. 2. (24). 1 𝑗 𝜕 (𝑎 (𝑡, 𝑧) 𝑋𝑝 (𝑡, 𝑧)) 𝑗!]𝑗 𝑧 𝑝 =. ∑ 𝑗0 +⋅⋅⋅+𝑗𝑝 =𝑗. 𝜕𝑧𝑗0 𝑎𝑝 (𝑡, 𝑧) 𝑗0. !]𝑗0. = ∑ 𝑙1 +𝑙2 =𝑗. 𝑝. Π𝑙=1. 𝜕𝑧𝑗𝑙 𝑋 (𝑡, 𝑧) , 𝑗𝑙 !]𝑗𝑙. 𝑑. +∑. 𝜕V𝑙11 𝑎 (V0 , V1 ) 𝑙1 !]𝑙1 ∑. 𝑝=0 𝑗0 +⋅⋅⋅+𝑗𝑝 =𝑗. (𝑙2 + 1) ]𝑢𝑙2 +1. 𝜕V𝑗10 𝑎𝑝 (V0 , V1 ) 𝑗0 !]𝑗0. 𝑝. Π𝑙=1 𝑢𝑗𝑙 ,. (27).

(6) 6. Abstract and Applied Analysis 𝐵𝑗 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼+1) ) = (𝑗 + 1) ]𝑢𝑗+1. 𝐵𝑗,𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼+1). (28). = for all 𝑗 ∈ 𝐼(𝛼), V0 , V1 ∈ 𝐷(0, 𝑅󸀠 ) and 𝑢ℎ ∈ C, ℎ ∈ 𝐼(𝛼). We define the sequences. sup. |V0 |<𝑅,|V1 |<𝑅,|𝑢ℎ. 󵄨󵄨 𝑛0 𝑛1 󵄨󵄨𝜕V 𝜕V Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ ℎ 󵄨 0 1 |<𝜌,ℎ∈𝐼(𝛼) 󵄨 × 𝐵𝑗 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼+1) )󵄨󵄨󵄨󵄨 (29). for all 𝑗 ∈ 𝐼(𝛼), all 𝑛0 , 𝑛1 ≥ 0, all 𝑙ℎ ≥ 0, ℎ ∈ 𝐼(𝛼 + 1), for all 𝛼 ≥ 0. We also recall the definition of the Kronecker symbol 𝛿0,𝑙 which is equal to 0 if 𝑙 ≠ 0 and equal to 1 if 𝑙 = 0.. 𝐴 𝑗,𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼+1) =. sup. |V0 |<𝑅,|V1 |<𝑅,|𝑢ℎ. 󵄨󵄨 𝑛0 𝑛1 󵄨󵄨𝜕V 𝜕V Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ ℎ 󵄨 0 1 |<𝜌,ℎ∈𝐼(𝛼). Proposition 3. The sequence 𝜑𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼) satisfies the following inequality:. 󵄨 × 𝐴 𝑗 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼+1) )󵄨󵄨󵄨󵄨 ,. 𝜑𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼) 𝛼!. ≤ ∑. ∑. 𝑘∈S. 𝛼1 +𝛼2 =𝛼. 𝛼2 ≥𝑆−𝑘. ×. +. 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! 𝑛0,1 +𝑛0,2 =𝑛0 ,𝑛1,1 +𝑛1,2 =𝑛1 𝑛0,1 !𝑛0,2 !𝑛1,1 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !𝑙ℎ,2 ! ∑. 𝑙ℎ,1 +𝑙ℎ,2 =𝑙ℎ ,ℎ∈𝐼(𝛼). 𝑏1,𝑘,𝛼1 ,𝑛0,1 ,𝑛1,1 ,𝑙0,1 𝛼1 !. Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ,1 ×. 2. 𝛼2 !. Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ,2. 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! 𝑛 !𝑛 !𝑛 !𝑛 !𝑛 !𝑛 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !𝑙ℎ,2 !𝑙ℎ,3 ! 𝑗∈𝐼(𝛼 +𝑘−𝑆) 𝑛0,1 +𝑛0,2 +𝑛0,3 =𝑛0 ,𝑛1,1 +𝑛1,2 +𝑛1,3 =𝑛1 0,1 0,2 0,3 1,1 1,2 1,3 ∑. ∑. 2. 𝑙ℎ,1 +𝑙ℎ,2 +𝑙ℎ,3 =𝑙ℎ ,ℎ∈𝐼(𝛼). ×. 𝑏1,𝑘,𝛼1 ,𝑛0,1 ,𝑛1,1 ,𝑙0,1 𝛼1 !. Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ,1 × 𝐴 𝑗,𝛼2 +𝑘−𝑆+1,𝑛0,2 ,𝑛1,2 ,(𝑙ℎ,2 )ℎ∈𝐼(𝛼 +𝑘−𝑆+1) 2. × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆+1) 𝛿0,𝑙ℎ,2 × + ∑ 𝑘∈S. ∑. 𝜑𝛼2 +𝑘−𝑆,𝑛0,3 ,𝑛1,3 ,(𝑙ℎ,3 )ℎ∈𝐼(𝛼 +𝑘−𝑆),ℎ ≠ 𝑗 ,𝑙𝑗,3 +1 2. 𝛼2 !. × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ,3. 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! 𝑛0,1 +𝑛0,2 =𝑛0 ,𝑛1,1 +𝑛1,2 =𝑛1 𝑛0,1 !𝑛0,2 !𝑛1,1 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !𝑙ℎ,2 ! ∑. 𝛼1 +𝛼2 =𝛼. 𝛼2 ≥𝑆−𝑘. ×. +. 𝜑𝛼2 +𝑘−𝑆,𝑛0,2 +1,𝑛1,2 ,(𝑙ℎ,2 )ℎ∈𝐼(𝛼 +𝑘−𝑆). 𝑙ℎ,1 +𝑙ℎ,2 =𝑙ℎ ,ℎ∈𝐼(𝛼). 𝑏2,𝑘,𝛼1 ,𝑛0,1 ,𝑛1,1 ,𝑙0,1 𝛼1 !. Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ,1 ×. 𝜑𝛼2 +𝑘−𝑆,𝑛0,2 ,𝑛1,2 +1,(𝑙ℎ,2 )ℎ∈𝐼(𝛼 +𝑘−𝑆) 2. 𝛼2 !. Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ,2. 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! 𝑛 !𝑛 !𝑛 !𝑛 !𝑛 !𝑛 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !𝑙ℎ,2 !𝑙ℎ,3 ! 𝑗∈𝐼(𝛼 +𝑘−𝑆) 𝑛0,1 +𝑛0,2 +𝑛0,3 =𝑛0 ,𝑛1,1 +𝑛1,2 +𝑛1,3 =𝑛1 0,1 0,2 0,3 1,1 1,2 1,3 ∑. ∑. 2. 𝑙ℎ,1 +𝑙ℎ,2 +𝑙ℎ,3 =𝑙ℎ ,ℎ∈𝐼(𝛼). ×. 𝑏2,𝑘,𝛼1 ,𝑛0,1 ,𝑛1,1 ,𝑙0,1 𝛼1 !. Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ,1 × 𝐵𝑗,𝛼2 +𝑘−𝑆+1,𝑛0,2 ,𝑛1,2 ,(𝑙ℎ,2 )ℎ∈𝐼(𝛼 +𝑘−𝑆+1). × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆+1) 𝛿0,𝑙ℎ,2 × + ∑ 𝑘∈S. ∑ 𝛼1 +𝛼2 =𝛼. 𝛼2 ≥𝑆−𝑘. 2. 𝜑𝛼2 +𝑘−𝑆,𝑛0,3 ,𝑛1,3 ,(𝑙ℎ,3 )ℎ∈𝐼(𝛼 +𝑘−𝑆),ℎ ≠ 𝑗 ,𝑙𝑗,3 +1 2. 𝛼2 !. 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! 𝑛 !𝑛 !𝑛 𝑛0,1 +𝑛0,2 =𝑛0 ,𝑛1,1 +𝑛1,2 =𝑛1 0,1 0,2 1,1 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !𝑙ℎ,2 ! ∑. 𝑙ℎ,1 +𝑙ℎ,2 =𝑙ℎ ,ℎ∈𝐼(𝛼). × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ,3.

(7) Abstract and Applied Analysis. 7 ×. +. 𝑏3,𝑘,𝛼1 ,𝑛0,1 ,𝑛1,1 ,𝑙0,1 𝛼1 !. Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ,1 ×. 𝜑𝛼2 +𝑘−𝑆,𝑛0,2 ,𝑛1,2 ,(𝑙ℎ,2 )ℎ∈𝐼(𝛼 +𝑘−𝑆) 2. 𝛼2 !. Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ,2. ̃𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼) 𝜔 𝛼! (30). for all 𝛼 ≥ 0, all 𝑛0 , 𝑛1 , 𝑙ℎ ≥ 0 for ℎ ∈ 𝐼(𝛼).. with. Proof. In order to get the inequality (30), we apply the differential operator 𝜕V𝑛00 𝜕V𝑛11 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎℎ 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. 𝜕V𝑛00,2 𝜕V𝑛11,2 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,2ℎ × (𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) )) = 𝜕V𝑛00,2 𝜕V𝑛11,2 Πℎ∈𝐼(𝛼2 +𝑘−𝑆) 𝜕𝑢𝑙ℎ,2ℎ. 𝜕V𝑛00 𝜕V𝑛11 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎℎ. × (𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) )) × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ,2 ,. × (𝑏3,𝑘,𝛼1 (V0 , V1 , 𝑢0 ) 𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) )) =. (33). 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! 𝑛0,1 +𝑛0,2 =𝑛0 ,𝑛1,1 +𝑛1,2 =𝑛1 𝑛0,1 !𝑛0,2 !𝑛1,1 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !𝑙ℎ,2 !. 𝜕V𝑛00,1 𝜕V𝑛11,1 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,1ℎ (𝑏1,𝑘,𝛼1 (V0 , V1 , 𝑢0 )). ∑. = 𝜕V𝑛00,1 𝜕V𝑛11,1 𝜕𝑢𝑙0,10 𝑏1,𝑘,𝛼1 (V0 , V1 , 𝑢0 ). 𝑙ℎ,1 +𝑙ℎ,2 =𝑙ℎ ,ℎ∈𝐼(𝛼). × 𝜕V𝑛00,1 𝜕V𝑛11,1 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,1ℎ (𝑏3,𝑘,𝛼1 (V0 , V1 , 𝑢0 )). (34). × Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ,1. × 𝜕V𝑛00,2 𝜕V𝑛11,2 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,2ℎ with. × (𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) )) , 𝜕V𝑛00 𝜕V𝑛11 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎℎ × (𝑏1,𝑘,𝛼1 (V0 , V1 , 𝑢0 ) 𝜕V0 𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) )) =. 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! 𝑛0,1 +𝑛0,2 =𝑛0 ,𝑛1,1 +𝑛1,2 =𝑛1 𝑛0,1 !𝑛0,2 !𝑛1,1 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !𝑙ℎ,2 ! ∑. × 𝜕V𝑛00,1 𝜕V𝑛11,1 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,1ℎ (𝑏1,𝑘,𝛼1 (V0 , V1 , 𝑢0 )). (35). × 𝜕𝑢𝑙ℎ,2ℎ (𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) )) × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ,2 .. × 𝜕V𝑛00,2 +1 𝜕V𝑛11,2 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,2ℎ (31). Moreover, we can write. 𝜕V𝑛00,1 𝜕V𝑛11,1 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,1ℎ. × (𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) )) = 𝜕V𝑛00,2 +1 𝜕V𝑛11,2 Πℎ∈𝐼(𝛼2 +𝑘−𝑆). 𝑙ℎ,1 +𝑙ℎ,2 =𝑙ℎ ,ℎ∈𝐼(𝛼). × (𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) )) .. 𝜕V𝑛00,2 +1 𝜕V𝑛11,2 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,2ℎ. By construction, we have. 𝐴 𝑗 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆+1) ) = ∑. (𝑏3,𝑘,𝛼1 (V0 , V1 , 𝑢0 )). = 𝜕V𝑛00,1 𝜕V𝑛11,1 𝜕𝑢𝑙0,10 𝑏3,𝑘,𝛼1 (V0 , V1 , 𝑢0 ) × Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ,1. 𝑙1 +𝑙2 =𝑗. (32). 𝑑. +∑. 𝜕V𝑙11 𝑎 (V0 , V1 ) 𝑙1 !]𝑙1 ∑. 𝑝=0 𝑗0 +⋅⋅⋅+𝑗𝑝 =𝑗. (𝑙2 + 1) ]𝑢𝑙2 +1. 𝜕V𝑗10 𝑎𝑝 (V0 , V1 ) 𝑗0 !]𝑗0. 𝑝. Π𝑙=1 𝑢𝑗𝑙. (36).

(8) 8. Abstract and Applied Analysis. for all 𝑗 ∈ 𝐼(𝛼2 + 𝑘 − 𝑆). Again, by the Leibniz formula, we get that. In the same way, one gets the following equalities: 𝜕V𝑛00 𝜕V𝑛11 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎℎ (𝑏2,𝑘,𝛼1 (V0 , V1 , 𝑢0 ) 𝜕V1. 𝜕V𝑛00 𝜕V𝑛11 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎℎ. × 𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) )). × (𝑏1,𝑘,𝛼1 (V0 , V1 , 𝑢0 ) 𝐴 𝑗 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆+1) ). =. ×𝜕𝑢𝑗 𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) )) =. ∑ 𝑛0,1 +𝑛0,2 +𝑛0,3 =𝑛0 ,𝑛1,1 +𝑛1,2 +𝑛1,3 =𝑛1. 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! 𝑛0,1 +𝑛0,2 =𝑛0 ,𝑛1,1 +𝑛1,2 =𝑛1 𝑛0,1 !𝑛0,2 !𝑛1,1 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !𝑙ℎ,2 ! ∑. × 𝜕V𝑛00,1 𝜕V𝑛11,1 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,1ℎ (𝑏2,𝑘,𝛼1 (V0 , V1 , 𝑢0 )). ( (𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ !). 𝑙ℎ,1 +𝑙ℎ,2 +𝑙ℎ,3 =𝑙ℎ ,ℎ∈𝐼(𝛼). × 𝜕V𝑛00,2 𝜕V𝑛11,2 +1 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,2ℎ. × (𝑛0,1 !𝑛0,2 !𝑛0,3 !𝑛1,1 !𝑛1,2 !𝑛1,3 !. × (𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) )). −1. × Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !𝑙ℎ,2 !𝑙ℎ,3 !) ) × 𝜕V𝑛00,1 𝜕V𝑛11,1 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,1ℎ. with the factorizations. × (𝑏1,𝑘,𝛼1 (V0 , V1 , 𝑢0 )). 𝜕V𝑛00,1 𝜕V𝑛11,1 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,1ℎ (𝑏2,𝑘,𝛼1 (V0 , V1 , 𝑢0 )). × 𝜕V𝑛00,2 𝜕V𝑛11,2 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,2ℎ. = 𝜕V𝑛00,1 𝜕V𝑛11,1 𝜕𝑢𝑙0,10 𝑏2,𝑘,𝛼1 (V0 , V1 , 𝑢0 ). × (𝐴 𝑗 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆+1) )) × 𝜕V𝑛00,3 𝜕V𝑛11,3. (40). 𝑙ℎ,1 +𝑙ℎ,2 =𝑙ℎ ,ℎ∈𝐼(𝛼). (41). × Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ,1 ,. 𝑙 +1 (Πℎ∈𝐼(𝛼),ℎ ≠ 𝑗 𝜕𝑢𝑙ℎ,3ℎ ) 𝜕𝑢𝑗,3𝑗. 𝜕V𝑛00,2 𝜕V𝑛11,2 +1 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,2ℎ. × 𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) ) . (37). × (𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) )) = 𝜕V𝑛00,2 𝜕V𝑛11,2 +1 Πℎ∈𝐼(𝛼2 +𝑘−𝑆) 𝜕𝑢𝑙ℎ,2ℎ. (42). × (𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) )). Inside the formula (37), we can rewrite the relations (34) and. × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ,2 . 𝜕V𝑛00,2 𝜕V𝑛11,2 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,2ℎ 𝐴 𝑗 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆+1) ) = 𝜕V𝑛00,2 𝜕V𝑛11,2 Πℎ∈𝐼(𝛼2 +𝑘−𝑆+1) 𝜕𝑢𝑙ℎ,2ℎ. We recall that (38). 𝐵𝑗 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆+1) ) = (𝑗 + 1) ]𝑢𝑗+1. × 𝐴 𝑗 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆+1) ) × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆+1) 𝛿0,𝑙ℎ,2. (43). for all 𝑗 ∈ 𝐼(𝛼2 + 𝑘 − 𝑆) and we deduce that 𝜕V𝑛00 𝜕V𝑛11 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎℎ (𝑏2,𝑘,𝛼1 (V0 , V1 , 𝑢0 ). with. × 𝐵𝑗 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆+1) ). 𝑙 +1. 𝜕V𝑛00,3 𝜕V𝑛11,3 (Πℎ∈𝐼(𝛼),ℎ ≠ 𝑗 𝜕𝑢𝑙ℎ,3ℎ ) 𝜕𝑢𝑗,3𝑗. ×𝜕𝑢𝑗 𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) )). × 𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) ) 𝑙 +1. = 𝜕V𝑛00,3 𝜕V𝑛11,3 (Πℎ∈𝐼(𝛼2 +𝑘−𝑆),ℎ ≠ 𝑗 𝜕𝑢𝑙ℎ,3ℎ ) 𝜕𝑢𝑗,3𝑗 × 𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) ) × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ,3 .. (39). =. ∑ 𝑛0,1 +𝑛0,2 +𝑛0,3 =𝑛0 ,𝑛1,1 +𝑛1,2 +𝑛1,3 =𝑛1. ( (𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ !). 𝑙ℎ,1 +𝑙ℎ,2 +𝑙ℎ,3 =𝑙ℎ ,ℎ∈𝐼(𝛼). × (𝑛0,1 !𝑛0,2 !𝑛0,3 !𝑛1,1 !𝑛1,2 !𝑛1,3 ! −1. × Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !𝑙ℎ,2 !𝑙ℎ,3 !) ).

(9) Abstract and Applied Analysis. 9. × 𝜕V𝑛00,1 𝜕V𝑛11,1 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,1ℎ. for 𝑚 = 1, 2, 3, all 𝑘 ∈ S, and. × (𝑏2,𝑘,𝛼1 (V0 , V1 , 𝑢0 )). A𝑗,𝛼 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ). × 𝜕V𝑛00,2 𝜕V𝑛11,2 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,2ℎ × (𝐵𝑗 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆+1) )) 𝑙 +1. × 𝜕V𝑛00,3 𝜕V𝑛11,3 (Πℎ∈𝐼(𝛼),ℎ ≠ 𝑗 𝜕𝑢𝑙ℎ,3ℎ ) 𝜕𝑢𝑗,3𝑗. × 𝜙𝛼2 +𝑘−𝑆 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆) ) . (44) Inside the formula (44), we can rewrite the relations (41) and 𝜕V𝑛00,2 𝜕V𝑛11,2 Πℎ∈𝐼(𝛼) 𝜕𝑢𝑙ℎ,2ℎ 𝐵𝑗 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆+1) ). 𝑛. 𝑙. 𝑛. 𝑈ℎ 𝑉0𝑉1 = 𝐴 𝑗,𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼) 0 1 Πℎ∈𝐼(𝛼) ℎ , ∑ 𝑛0 ! 𝑛1 ! 𝑙ℎ ! 𝑛 ,𝑛 ,𝑙 ≥0,ℎ∈𝐼(𝛼) 0. 1 ℎ. (48) B𝑗,𝛼 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ) 𝑛. =. ∑ 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼). 𝐵𝑗,𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼). 𝑙. 𝑛. 𝑈ℎ 𝑉0 0 𝑉1 1 Πℎ∈𝐼(𝛼) ℎ 𝑛0 ! 𝑛1 ! 𝑙ℎ !. for all 𝛼 ≥ 0, all 𝑗 ∈ 𝐼(𝛼). We also introduce the following linear operators acting on G[[𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊]]. Let. = 𝜕V𝑛00,2 𝜕V𝑛11,2 Πℎ∈𝐼(𝛼2 +𝑘−𝑆+1) 𝜕𝑢𝑙ℎ,2ℎ. (45). × 𝐵𝑗 (V0 , V1 , (𝑢ℎ )ℎ∈𝐼(𝛼2 +𝑘−𝑆+1) ). DA Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊). × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆+1) 𝛿0,𝑙ℎ,2 = ∑ ( ∑ A𝑗,𝛼+1 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼+1) ). with the factorization (39).. 𝛼≥0. 𝑗∈𝐼(𝛼). 2.4. Majorant Series and a Functional Equation with Infinitely Many Variables Definition 4. One denotes by G[[𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊]] the vector space of formal series in the variables 𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , and 𝑊 of the form Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊). × (𝜕𝑈𝑗 Ψ𝛼 ) (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ) ). (49) DB Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊) = ∑ ( ∑ B𝑗,𝛼+1 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼+1) ). = ∑ Ψ𝛼 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ) 𝛼≥0. 𝛼. (46). 𝑊 , 𝛼!. 𝛼≥0. 𝑗∈𝐼(𝛼). × (𝜕𝑈𝑗 Ψ𝛼 ) (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ) ). where Ψ𝛼 ∈ C[[𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ]] for all 𝛼 ≥ 0. We keep the notations of the previous section and we introduce the following formal series: 𝐵𝑚,𝑘 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊) 𝑛. = ∑ ( ∑ 𝑏𝑚,𝑘,𝛼,𝑛0 ,𝑛1 ,𝑙0 𝛼≥0. 𝑛0 ,𝑛1 ,𝑙0 ≥0. 𝑛. 𝑛. 𝑛. 𝑛. 𝑙. 1 ℎ. 𝑊𝛼 𝛼!. for all Ψ ∈ G[[𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊]]. We stress the fact that although these operators act on G[[𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊]] their image does not have to belong to this space.. Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊). 𝑈ℎ 𝑉0𝑉1 ̃𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼) 0 1 Πℎ∈𝐼(𝛼) ℎ ) = ∑( 𝜔 ∑ 𝑛0 ! 𝑛1 ! 𝑙ℎ ! 𝛼≥0 𝑛 ,𝑛 ,𝑙 ≥0,ℎ∈𝐼(𝛼) ×. 𝑊𝛼 𝛼!. Proposition 5. A formal series. 𝑙. 𝑉0 0 𝑉1 1 𝑈00 𝑊𝛼 ) , 𝑛0 ! 𝑛1 ! 𝑙0 ! 𝛼!. ̃ (𝑉0 , 𝑉1 , (𝑈ℎ ) , 𝑊) Ω ℎ≥0. 0. 𝑊𝛼 , 𝛼!. = ∑( 𝛼≥0. × (47). ∑. 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼). 𝑊𝛼 𝛼!. 𝜓𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼). 𝑛. 𝑙. 𝑈ℎ 𝑉0 0 𝑉1 1 Πℎ∈𝐼(𝛼) ℎ ) 𝑛0 ! 𝑛1 ! 𝑙ℎ !. (50).

(10) 10. Abstract and Applied Analysis −𝑆+𝑘 + 𝐵2,𝑘 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊) 𝜕𝑊. satisfies the following functional equation:. × DB Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊)). Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊). −𝑆+𝑘 Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊) + ∑ 𝐵3,𝑘 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊) 𝜕𝑊. −𝑆+𝑘 𝜕𝑉0 Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊) = ∑ (𝐵1,𝑘 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊) 𝜕𝑊. 𝑘∈S. 𝑘∈S. ̃ (𝑉0 , 𝑉1 , (𝑈ℎ ) , 𝑊) +Ω ℎ≥0. −𝑆+𝑘 + 𝐵1,𝑘 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊) 𝜕𝑊. (51). × DA Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊)) −𝑆+𝑘 𝜕𝑉1 Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊) + ∑ (𝐵2,𝑘 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊) 𝜕𝑊. if and only if its coefficients 𝜓𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼) satisfy the following recursion:. 𝑘∈S. 𝜓𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼) 𝛼!. = ∑. ∑. 𝑘∈S. 𝛼1 +𝛼2 =𝛼. 𝛼2 ≥𝑆−𝑘. ×. +. 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! 𝑛0,1 +𝑛0,2 =𝑛0 ,𝑛1,1 +𝑛1,2 =𝑛1 𝑛0,1 !𝑛0,2 !𝑛1,1 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !𝑙ℎ,2 ! ∑. 𝑙ℎ,1 +𝑙ℎ,2 =𝑙ℎ ,ℎ∈𝐼(𝛼). 𝑏1,𝑘,𝛼1 ,𝑛0,1 ,𝑛1,1 ,𝑙0,1 𝛼1 !. Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ,1 ×. 𝜓𝛼2 +𝑘−𝑆,𝑛0,2 +1,𝑛1,2 ,(𝑙ℎ,2 )ℎ∈𝐼(𝛼 +𝑘−𝑆) 2. 𝛼2 !. Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ,2. 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! 𝑛 !𝑛 !𝑛 !𝑛 !𝑛 !𝑛 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !𝑙ℎ,2 !𝑙ℎ,3 ! 𝑗∈𝐼(𝛼 +𝑘−𝑆) 𝑛0,1 +𝑛0,2 +𝑛0,3 =𝑛0 ,𝑛1,1 +𝑛1,2 +𝑛1,3 =𝑛1 0,1 0,2 0,3 1,1 1,2 1,3 ∑. ∑. 2. 𝑙ℎ,1 +𝑙ℎ,2 +𝑙ℎ,3 =𝑙ℎ ,ℎ∈𝐼(𝛼). ×. 𝑏1,𝑘,𝛼1 ,𝑛0,1 ,𝑛1,1 ,𝑙0,1 𝛼1 !. Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ,1 × 𝐴 𝑗,𝛼2 +𝑘−𝑆+1,𝑛0,2 ,𝑛1,2 ,(𝑙ℎ,2 )ℎ∈𝐼(𝛼 +𝑘−𝑆+1) 2. × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆+1) 𝛿0,𝑙ℎ,2 × + ∑ 𝑘∈S. ∑ 𝛼1 +𝛼2 =𝛼. 𝛼2 ≥𝑆−𝑘. ×. +. 𝑙ℎ,1 +𝑙ℎ,2 =𝑙ℎ ,ℎ∈𝐼(𝛼). Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ,1 ×. 𝜓𝛼2 +𝑘−𝑆,𝑛0,2 ,𝑛1,2 +1,(𝑙ℎ,2 )ℎ∈𝐼(𝛼 +𝑘−𝑆) 2. 𝛼2 !. Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ,2. 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! 𝑛0,1 +𝑛0,2 +𝑛0,3 =𝑛0 ,𝑛1,1 +𝑛1,2 +𝑛1,3 =𝑛1 𝑛0,1 !𝑛0,2 !𝑛0,3 !𝑛1,1 !𝑛1,2 !𝑛1,3 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !𝑙ℎ,2 !𝑙ℎ,3 ! ∑. 𝑙ℎ,1 +𝑙ℎ,2 +𝑙ℎ,3 =𝑙ℎ ,ℎ∈𝐼(𝛼). 𝑏2,𝑘,𝛼1 ,𝑛0,1 ,𝑛1,1 ,𝑙0,1 𝛼1 !. Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ,1 × 𝐵𝑗,𝛼2 +𝑘−𝑆+1,𝑛0,2 ,𝑛1,2 ,(𝑙ℎ,2 )ℎ∈𝐼(𝛼 +𝑘−𝑆+1) 2. × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆+1) 𝛿0,𝑙ℎ,2 ×. 𝑘∈S. × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ,3. 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! 𝑛 !𝑛 !𝑛 𝑛0,1 +𝑛0,2 =𝑛0 ,𝑛1,1 +𝑛1,2 =𝑛1 0,1 0,2 1,1 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !𝑙ℎ,2 !. 𝛼1 !. 𝑗∈𝐼(𝛼2 +𝑘−𝑆). + ∑. 2. 𝛼2 !. ∑. 𝑏2,𝑘,𝛼1 ,𝑛0,1 ,𝑛1,1 ,𝑙0,1. ∑. ×. 𝜓𝛼2 +𝑘−𝑆,𝑛0,3 ,𝑛1,3 ,(𝑙ℎ,3 )ℎ∈𝐼(𝛼 +𝑘−𝑆),ℎ ≠ 𝑗 ,𝑙𝑗,3 +1. ∑ 𝛼1 +𝛼2 =𝛼. 𝛼2 ≥𝑆−𝑘. 𝜓𝛼2 +𝑘−𝑆,𝑛0,3 ,𝑛1,3 ,(𝑙ℎ,3 )ℎ∈𝐼(𝛼 +𝑘−𝑆),ℎ ≠ 𝑗 ,𝑙𝑗,3 +1 2. 𝛼2 !. n0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! 𝑛 !𝑛 !𝑛 𝑛0,1 +𝑛0,2 =𝑛0 ,𝑛1,1 +𝑛1,2 =𝑛1 0,1 0,2 1,1 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !𝑙ℎ,2 ! ∑. 𝑙ℎ,1 +𝑙ℎ,2 =𝑙ℎ ,ℎ∈𝐼(𝛼). × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ,3.

(11) Abstract and Applied Analysis. 11 ×. 𝑏3,𝑘,𝛼1 ,𝑛0,1 ,𝑛1,1 ,𝑙0,1 𝛼1 !. Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ,1 ×. 𝜓𝛼2 +𝑘−𝑆,𝑛0,2 ,𝑛1,2 ,(𝑙ℎ,2 )ℎ∈𝐼(𝛼 +𝑘−𝑆) 2. 𝛼2 !. Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ,2. ̃𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼) 𝜔. +. 𝛼! (52). for all 𝛼 ≥ 0, all 𝑛0 , 𝑛1 , 𝑙ℎ ≥ 0 with ℎ ∈ 𝐼(𝛼). × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ,2 ). Proof. We proceed by identification of the coefficients in the Taylor expansion with respect to the variables 𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) , and 𝑊 for all 𝛼 ≥ 0. By definition, we have that. 𝑛. 𝑙. 𝑛. × 𝑉0 0 𝑉1 1 Πℎ∈𝐼(𝛼) 𝑈ℎℎ . (55). −𝑆+𝑘 𝜕𝑉0 Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊) 𝐵1,𝑘 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊) 𝜕𝑊. = ∑ ∑ C1𝛼1 ,𝛼2 𝑊𝛼 , 𝛼≥0. (53). 𝛼1 +𝛼2 =𝛼. We also have that −𝑆+𝑘 DA Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊) 𝐵1,𝑘 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊) 𝜕𝑊. 𝛼2 ≥𝑆−𝑘. = ∑ ∑ F1𝛼1 ,𝛼2 𝑊𝛼 ,. C1𝛼1 ,𝛼2. can be rewritten, using the where the coefficients Kronecker symbol 𝛿0,𝑚 , in the form. 𝛼1 +𝛼2 =𝛼. 𝛼2 ≥𝑆−𝑘. where the coefficients F1𝛼1 ,𝛼2 can be rewritten in the form. C1𝛼1 ,𝛼2 =(. 𝛼≥0. (56). F1𝛼1 ,𝛼2. 𝑏1,𝑘,𝛼1 ,𝑛0 ,𝑛1 ,𝑙0. ∑. 𝛼1 !. 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼). 𝑛. × Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ ×(. =. 𝑙. 𝑛. 𝑈ℎ 𝑉0 0 𝑉1 1 Πℎ∈𝐼(𝛼) ℎ ) 𝑛0 ! 𝑛1 ! 𝑙ℎ !. ∑. 𝑗∈𝐼(𝛼2 −𝑆+𝑘). (. × Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ 𝑛. 2. ×. 𝛼2 !. 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼). 𝑛. 𝛼1 !. 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼). 𝜓𝛼2 +𝑘−𝑆,𝑛0 +1,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼 +𝑘−𝑆). ∑. 𝑏1,𝑘,𝛼1 ,𝑛0 ,𝑛1 ,𝑙0. ∑. 𝑙. 𝑛. 𝑈ℎ 𝑉0 0 𝑉1 1 Πℎ∈𝐼(𝛼) ℎ ) 𝑛0 ! 𝑛1 ! 𝑙ℎ !. 𝑙. 𝑛. 𝑈ℎ 𝑉0𝑉1 × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ 0 1 Πℎ∈𝐼(𝛼) ℎ ) . 𝑛0 ! 𝑛1 ! 𝑙ℎ ! (54). ×(. ∑ 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼). 𝐴 𝑗,𝛼2 −𝑆+𝑘+1,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼 −𝑆+𝑘+1) 2. × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 −𝑆+𝑘+1) 𝛿0,𝑙ℎ Hence,. 𝑛. C1𝛼1 ,𝛼2 =. 𝑙. 𝑛. 𝑈ℎ 𝑉0𝑉1 × 0 1 Πℎ∈𝐼(𝛼) ℎ ) 𝑛0 ! 𝑛1 ! 𝑙ℎ !. ∑ 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼). (. 𝑏1,𝑘,𝛼1 ,𝑛0,1 ,𝑛1,1 ,𝑙0,1. ∑ 𝑛0,1 +𝑛0,2 =𝑛0 ,𝑛1,1 +𝑛1,2 =𝑛1. 𝑙ℎ,1 +𝑙ℎ,2 =𝑙ℎ ,ℎ∈𝐼(𝛼). 𝛼1 !𝑛0,1 !𝑛1,1 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !. × Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ,1 ×. ×(. 𝜓𝛼2 −𝑆+𝑘,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼 −𝑆+𝑘),ℎ ≠ 𝑗 ,𝑙𝑗 +1. ∑. 2. 𝛼2 !. 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼). × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 −𝑆+𝑘) 𝛿0,𝑙ℎ 𝑛. 𝜓𝛼2 +𝑘−𝑆,𝑛0,2 +1,𝑛1,2 ,(𝑙ℎ,2 )ℎ∈𝐼(𝛼 +𝑘−𝑆) 2. 𝛼2 !𝑛0,2 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,2 !. ×. 𝑛. 𝑙. 𝑈ℎ 𝑉0 0 𝑉1 1 Πℎ∈𝐼(𝛼) ℎ ) . 𝑛0 ! 𝑛1 ! 𝑙ℎ ! (57).

(12) 12. Abstract and Applied Analysis. Therefore,. F1𝛼1 ,𝛼2 =. (. ∑ 𝑗∈𝐼(𝛼2 −𝑆+𝑘). (. ∑ 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼). 𝑏1,𝑘,𝛼1 ,𝑛0,1 ,𝑛1,1 ,𝑙0,1. ∑ 𝑛0,1 +𝑛0,2 +𝑛0,3 =𝑛0 ,𝑛1,1 +𝑛1,2 +𝑛1,3 =𝑛1. 𝑙ℎ,1 +𝑙ℎ,2 +𝑙ℎ,3 =𝑙ℎ ,ℎ∈𝐼(𝛼). ×. ×. 𝑛. 𝛼1 !𝑛0,1 !𝑛1,1 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !. Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ,1. 𝐴 𝑗,𝛼2 −𝑆+𝑘+1,𝑛0,2 ,𝑛1,2 ,(𝑙ℎ,2 )ℎ∈𝐼(𝛼 −𝑆+𝑘+1) 2. 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 )ℎ∈𝐼(𝛼 −𝑆+𝑘),ℎ ≠ 𝑗 ,𝑙𝑗,3 +1 2. 𝛼2 !𝑛0,3 !𝑛1,3 !Πℎ∈𝐼(𝛼) 𝑙ℎ,3 !. (58) Πℎ∈𝐼(𝛼)\𝐼(𝛼2 −𝑆+𝑘) 𝛿0,𝑙ℎ,3 ). 𝑙. 𝑛. ×𝑉0 0 𝑉1 1 Πℎ∈𝐼(𝛼) 𝑈ℎℎ ) .. On the other hand, using similar computations we get. ×. 𝜓𝛼2 +𝑘−𝑆,𝑛0,2 ,𝑛1,2 +1,(𝑙ℎ,2 )ℎ∈𝐼(𝛼 +𝑘−𝑆) 2. 𝛼2 !𝑛0,2 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,2 !. −𝑆+𝑘 𝜕𝑉1 Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊) 𝐵2,𝑘 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊) 𝜕𝑊. (59). = ∑ ∑ C2𝛼1 ,𝛼2 𝑊𝛼 , 𝛼≥0. × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ,2 ). 𝛼1 +𝛼2 =𝛼 𝛼2 ≥𝑆−𝑘. 𝑛. (60). C2𝛼1 ,𝛼2. =. 𝑙. 𝑛. × 𝑉0 0 𝑉1 1 Πℎ∈𝐼(𝛼) 𝑈ℎℎ .. where. We also have that (. ∑ 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼). −𝑆+𝑘 𝐵2,𝑘 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊) 𝜕𝑊 DB Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊). 𝑏2,𝑘,𝛼1 ,𝑛0,1 ,𝑛1,1 ,𝑙0,1. ∑ 𝑛0,1 +𝑛0,2 =𝑛0 ,𝑛1,1 +𝑛1,2 =𝑛1. 𝑙ℎ,1 +𝑙ℎ,2 =𝑙ℎ ,ℎ∈𝐼(𝛼). × Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ,1. F2𝛼1 ,𝛼2 =. ∑ 𝑗∈𝐼(𝛼2 −𝑆+𝑘). (. = ∑ ∑ F2𝛼1 ,𝛼2 𝑊𝛼 ,. 𝛼1 !𝑛0,1 !𝑛1,1 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !. ∑ 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼). (. 𝛼≥0. 𝛼1 +𝛼2 =𝛼 𝛼2 ≥𝑆−𝑘. where. 𝑏2,𝑘,𝛼1 ,𝑛0,1 ,𝑛1,1 ,𝑙0,1. ∑ 𝑛0,1 +𝑛0,2 +𝑛0,3 =𝑛0 ,𝑛1,1 +𝑛1,2 +𝑛1,3 =𝑛1. 𝑙ℎ,1 +𝑙ℎ,2 +𝑙ℎ,3 =𝑙ℎ ,ℎ∈𝐼(𝛼). 𝛼1 !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 )ℎ∈𝐼(𝛼 −𝑆+𝑘+1) 2. 𝑛0,2 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,2 !. Πℎ∈𝐼(𝛼)\𝐼(𝛼2 −𝑆+𝑘+1) 𝛿0,𝑙ℎ,2. (61).

(13) Abstract and Applied Analysis. 13. ×. 𝜓𝛼2 −𝑆+𝑘,𝑛0,3 ,𝑛1,3 ,(𝑙ℎ,3 )ℎ∈𝐼(𝛼 −𝑆+𝑘),ℎ ≠ 𝑗 ,𝑙𝑗,3 +1 2. 𝛼2 !𝑛0,3 !𝑛1,3 !Πℎ∈𝐼(𝛼) 𝑙ℎ,3 !. Πℎ∈𝐼(𝛼)\𝐼(𝛼2 −𝑆+𝑘) 𝛿0,𝑙ℎ,3 ) (62). 𝑛. 𝑙. 𝑛. ×𝑉0 0 𝑉1 1 Πℎ∈𝐼(𝛼) 𝑈ℎℎ ) , −𝑆+𝑘 𝐵3,𝑘 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊) 𝜕𝑊 Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊) = ∑ ∑ C3𝛼1 ,𝛼2 𝑊𝛼 , 𝛼≥0. 3. Convergent Series Solutions for a Functional Equation with Infinitely Many Variables. where C3𝛼1 ,𝛼2 =. (63). 𝛼1 +𝛼2 =𝛼. 𝛼2 ≥𝑆−𝑘. (. ∑ 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼). 𝑏3,𝑘,𝛼1 ,𝑛0,1 ,𝑛1,1 ,𝑙0,1. ∑ 𝑛0,1 +𝑛0,2 =𝑛0 ,𝑛1,1 +𝑛1,2 =𝑛1. 𝑙ℎ,1 +𝑙ℎ,2 =𝑙ℎ ,ℎ∈𝐼(𝛼). 𝛼1 !𝑛0,1 !𝑛1,1 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !. × Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ,1 ×. Definition 7. Let 𝛼 ≥ 0 be an integer. One denotes by 𝐸𝜌,𝛼,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼) the vector space of formal series. 𝜓𝛼2 +𝑘−𝑆,𝑛0,2 ,𝑛1,2 ,(𝑙ℎ,2 )ℎ∈𝐼(𝛼 +𝑘−𝑆) 2. 𝛼2 !𝑛0,2 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,2 ! Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ). × Πℎ∈𝐼(𝛼)\𝐼(𝛼2 +𝑘−𝑆) 𝛿0,𝑙ℎ,2 ) 𝑛. 𝑛. 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 𝑟𝑏 (𝛼) = ∑𝛼𝑛=0 1/(𝑛 + 1)𝑏 for all 𝛼 ≥ 0 and 𝑈ℎ = 𝛿/(ℎ𝑏 + 1) for all ℎ ≥ 0.. 𝑛. 𝑙. × 𝑉0 0 𝑉1 1 Πℎ∈𝐼(𝛼) 𝑈ℎℎ .. 𝑛. 𝑉0𝑉1 = 𝜓𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼) 0 1 ∑ 𝑛0 ! 𝑛1 ! 𝑛 ,𝑛 ,𝑙 ≥0,ℎ∈𝐼(𝛼) 0. 1 ℎ. (67). 𝑙. 𝑈ℎ × Πℎ∈𝐼(𝛼) ℎ 𝑙ℎ !. (64). Finally, gathering the expansions (55), (58), (60), and (62) with (64) yields the result. Proposition 6. The sequences 𝜑𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼) 𝜓𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼) satisfy the following inequalities: 𝜑𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼) ≤ 𝜓𝛼,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼). and. (65). for all 𝛼 ≥ 0, all 𝑛0 , 𝑛1 ≥ 0, all 𝑙ℎ ≥ 0, ℎ ∈ 𝐼(𝛼). Proof. For 𝛼 = 0, using the recursions (16) and (52), we get that. that belong to C[[𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ]] such that the series. 󵄩󵄩 󵄩 󵄩󵄩Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) )󵄩󵄩󵄩 󵄩 󵄩𝜌,𝛼,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼) 󵄨󵄨󵄨𝜓 󵄨󵄨 󵄨󵄨 𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼) 󵄨󵄨󵄨 = ∑ exp (𝜎𝑟𝑏 (𝛼) 𝜌) 𝑛 ,𝑛 ,𝑙 ≥0,ℎ∈𝐼(𝛼) 0. 𝑛. × ̃0,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(0) = 𝜓0,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(0) 𝜑0,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(0) = 𝑤. (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.. (68). 1 ℎ. 𝑛. 𝑙. 𝑉00 𝑉11 Πℎ∈𝐼(𝛼) 𝑈ℎℎ (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ + 𝛼)!. is convergent. One denotes also by 𝐺(𝜌,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ≥0 ,𝑊) the vector space of formal series.

(14) 14. Abstract and Applied Analysis Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊) (69). 𝑊𝛼 , 𝛼!. = ∑ Ψ𝛼 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ) 𝛼≥0. Let Ψ(𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ) belong to 𝐸𝜌,𝛼,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼) . Then, the following inequality: 󵄩󵄩 󵄩󵄩𝑏 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ) 󵄩 󵄩 × Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) )󵄩󵄩󵄩󵄩𝜌,𝛼,𝑉 ,𝑉 ,(𝑈 ) 0. ≤ |𝑏| (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ). where Ψ𝛼 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ) belong to 𝐸𝜌,𝛼,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼) for all 𝛼 ≥ 0, such that the series. 1. ℎ ℎ∈𝐼(𝛼). 󵄩 󵄩 × 󵄩󵄩󵄩󵄩Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) )󵄩󵄩󵄩󵄩𝜌,𝛼,𝑉 ,𝑉 ,(𝑈 ) 0. 1. ℎ ℎ∈𝐼(𝛼). (73) 󵄩󵄩 󵄩 󵄩󵄩Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊)󵄩󵄩󵄩 󵄩 󵄩(𝜌,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ≥0 ,𝑊). holds. (70). 𝛼 󵄩 󵄩 = ∑ 󵄩󵄩󵄩Ψ𝛼 󵄩󵄩󵄩𝜌,𝛼,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼) 𝑊. Proof. Let Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ). 𝛼≥0. 𝑛. 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 𝐸𝜌,𝛼,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼) constructed above.. ∑ 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼). 𝜓𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼). 𝑛. 𝑉0 0 𝑉1 1 𝑛0 ! 𝑛1 !. (74). 𝑙. 𝑈ℎ × Πℎ∈𝐼(𝛼) ℎ 𝑙ℎ ! which belongs to 𝐸𝜌,𝛼,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼) . By definition, we have that 󵄩󵄩 󵄩󵄩𝑏 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ) 󵄩. Proposition 8. Consider a formal series. 󵄩 × Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) )󵄩󵄩󵄩󵄩𝜌,𝛼,𝑉 ,𝑉 ,(𝑈 ) 0. 𝑏 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ) 𝑛. =. =. ∑ 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼). 𝑏𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼). 𝑛. 𝑉0 0 𝑉1 1 𝑛0 ! 𝑛1 !. (71). 1. ℎ ℎ∈𝐼(𝛼). 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 = ( (𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ !) ∑ ∑ 󵄨󵄨 󵄨 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼) 󵄨󵄨󵄨𝑛0,1 +𝑛0,2 =𝑛0 ,𝑛1,1 +𝑛1,2 =𝑛1 󵄨󵄨 𝑙ℎ,1 +𝑙ℎ,2 =𝑙ℎ ,ℎ∈𝐼(𝛼) × (𝑛0,1 !𝑛0,2 !𝑛1,1 !𝑛1,2 !. 𝑙. 𝑈ℎ × Πℎ∈𝐼(𝛼) ℎ 𝑙ℎ !. −1. × Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !𝑙ℎ,2 !) ) × 𝑏𝑛0,1 ,𝑛1,1 ,(𝑙ℎ,1 )ℎ∈𝐼(𝛼) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 × 𝜓𝑛0,2 ,𝑛1,2 ,(𝑙ℎ,2 )ℎ∈𝐼(𝛼) 󵄨󵄨󵄨 󵄨󵄨󵄨 󵄨󵄨 󵄨. which is absolutely convergent on the polydisc 𝐷(0, 𝑉0 ) × 𝐷(0, 𝑉1 )×ℎ∈𝐼(𝛼) 𝐷(0, 𝑈ℎ ). One uses the notation. |𝑏| (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ). × 𝑛0. =. 𝑛0 ,𝑛1 ,𝑙ℎ. 𝑛. 1 󵄨󵄨 󵄨𝑉 𝑉 󵄨󵄨𝑏𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼) 󵄨󵄨󵄨 0 1 󵄨 󵄨 𝑛0 ! 𝑛1 ! ≥0,ℎ∈𝐼(𝛼). ∑. 𝑙. 𝑈ℎ × Πℎ∈𝐼(𝛼) ℎ . 𝑙ℎ !. 1 exp (𝜎𝑟𝑏 (𝛼) 𝜌) 𝑛. (72). ×. 𝑛. 𝑙. 𝑉00 𝑉11 Πℎ∈𝐼(𝛼) 𝑈ℎℎ (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ + 𝛼)!. . (75). We can give upper bounds for this latter expression.

(15) Abstract and Applied Analysis. 15. 󵄩󵄩 󵄩 󵄩󵄩𝑏 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ) Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) )󵄩󵄩󵄩 󵄩 󵄩𝜌,𝛼,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼) ≤. ∑. ∑. 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼) 𝑛0,1 +𝑛0,2 =𝑛0 ,𝑛1,1 +𝑛1,2 =𝑛1 𝑙ℎ,1 +𝑙ℎ,2 =𝑙ℎ ,ℎ∈𝐼(𝛼). ×. (. (𝑛0,2 + 𝑛1,2 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ,2 + 𝛼)! 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! × ) 𝑛0,2 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,2 ! (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ + 𝛼)!. 󵄨󵄨 󵄨 𝑛 𝑛 𝑙 󵄨󵄨𝑏𝑛0,1 ,𝑛1,1 ,(𝑙ℎ,1 )ℎ∈𝐼(𝛼) 󵄨󵄨󵄨 𝑉00,2 𝑉11,2 Πℎ∈𝐼(𝛼) 𝑈ℎℎ,2 𝑙ℎ,1 1 󵄨󵄨 󵄨󵄨 󵄨 󵄨 𝑉𝑛0,1 𝑉𝑛1,1 Π 󵄨 󵄨 𝜓 . 𝑈 × 󵄨󵄨 𝑛0,2 ,𝑛1,2 ,(𝑙ℎ,2 )ℎ∈𝐼(𝛼) 󵄨󵄨 𝑛0,1 !𝑛1,1 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 ! 0 1 ℎ∈𝐼(𝛼) ℎ exp (𝜎𝑟𝑏 (𝛼) 𝜌) (𝑛0,2 + 𝑛1,2 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ,2 + 𝛼)! (76). Lemma 9. For all integers 𝛼, 𝑛0 , 𝑛1 ≥ 0, all 𝑙ℎ ≥ 0, all 0 ≤ 𝑛0,2 ≤ 𝑛0 , all 0 ≤ 𝑛1,2 ≤ 𝑛1 , and all 0 ≤ 𝑙ℎ,2 ≤ 𝑙ℎ for ℎ ∈ 𝐼(𝛼), one has that 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! (𝑛0,2 + 𝑛1,2 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ,2 + 𝛼)! ≤ 1. 𝑛0,2 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,2 ! (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ + 𝛼)!. 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 󵄩󵄩 󵄩󵄩 󵄩󵄩𝜕𝑈 Ψ (𝑉0 , 𝑉1 , (𝑈ℎ ) 󵄩 󵄩󵄩 𝑗 󵄩󵄩𝜌,𝛼,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼) ℎ∈𝐼(𝛼󸀠 ) )󵄩. (77) ≤. Proof. For any integers 𝑎 ≤ 𝑏 and 𝛼 ≥ 0, one has (𝑎 + 𝛼)! 𝑎! ≤ (𝑏 + 𝛼)! 𝑏!. 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! (𝑛0,2 + 𝑛1,2 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ,2 + 𝛼)! 𝑛0,2 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,2 ! (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ + 𝛼)! 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! (𝑛0,2 + 𝑛1,2 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ,2 )! ≤ . 𝑛0,2 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,2 ! (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ )!. 0. ,. 󵄩󵄩 󵄩 󵄩󵄩Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼󸀠 ) )󵄩󵄩󵄩 󵄩 󵄩𝜌,𝛼,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼) ≤. exp (−𝜎𝜌 ((𝛼 − 𝛼󸀠 ) /(𝛼 + 1)𝑏 )). (84). 󸀠. Π𝛼−𝛼 𝑙=1 (𝛼 − 𝑙 + 1) 0. for all 𝑛0,1 + 𝑛0,2 = 𝑛0 , 𝑛1,1 + 𝑛1,2 = 𝑛1 , 𝑙ℎ,1 + 𝑙ℎ,2 = 𝑙ℎ . Therefore, we deduce that 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! (𝑛0,2 + 𝑛1,2 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ,2 + 𝛼)! 𝑛0,2 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,2 ! (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ + 𝛼)! (𝑛0,1 + 𝑛1,1 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ,1 )!. ℎ ℎ∈𝐼(𝛼󸀠 ). 1. 󵄩 󵄩 × 󵄩󵄩󵄩󵄩Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼󸀠 ) )󵄩󵄩󵄩󵄩𝜌,𝛼󸀠 ,𝑉 ,𝑉 ,(𝑈 ). (𝑛0,1 + 𝑛1,1 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ,1 )! (𝑛0,2 + 𝑛1,2 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ,2 )! (80). 𝑛0,1 !𝑛1,1 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !. (83). 󸀠. 𝛼−𝛼 −1 𝑉𝑘 Π𝑙=1 (𝛼 − 𝑙 + 1). 0. (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ )!. ≤. ,. exp (−𝜎𝜌 ((𝛼 − 𝛼󸀠 ) /(𝛼 + 1)𝑏 )) 󵄩 󵄩 × 󵄩󵄩󵄩󵄩Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼󸀠 ) )󵄩󵄩󵄩󵄩𝜌,𝛼󸀠 ,𝑉 ,𝑉 ,(𝑈 ). (79). 𝑛0 !𝑛1 !Πℎ∈𝐼(𝛼) 𝑙ℎ ! 𝑛0,1 !𝑛0,2 !𝑛1,1 !𝑛1,2 !Πℎ∈𝐼(𝛼) 𝑙ℎ,1 !𝑙ℎ,2 ! ≤. ℎ ℎ∈𝐼(𝛼󸀠 ). 1. 󵄩󵄩 󵄩 󵄩󵄩𝜕𝑉𝑘 Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼󸀠 ) )󵄩󵄩󵄩 󵄩 󵄩𝜌,𝛼,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼) ≤. Now, from the identity (𝐴 + 𝐵)𝑛0 +𝑛1 +∑ℎ∈𝐼(𝛼) 𝑙ℎ = (𝐴 + 𝐵)𝑛0 (𝐴 + 𝐵)𝑛1 × Πℎ∈𝐼(𝛼) (𝐴 + 𝐵)𝑙ℎ and the binomial formula, we deduce that. (82). 󸀠. 𝛼−𝛼 −1 𝑈𝑗 Π𝑙=1 (𝛼 − 𝑙 + 1). 󵄩 󵄩 × 󵄩󵄩󵄩󵄩Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼󸀠 ) )󵄩󵄩󵄩󵄩𝜌,𝛼󸀠 ,𝑉 ,𝑉 ,(𝑈 ). (78). by using the factorization (𝑎 + 𝛼)! = (𝑎 + 𝛼)(𝑎 + 𝛼 − 1) ⋅ ⋅ ⋅ (𝑎 + 1)𝑎!. Therefore, one gets the inequality. exp (−𝜎𝜌 ((𝛼 − 𝛼󸀠 ) /(𝛼 + 1)𝑏 )). ℎ ℎ∈𝐼(𝛼󸀠 ). 1. for all Ψ(𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼󸀠 ) ) ∈ 𝐸𝜌,𝛼󸀠 ,𝑉0 ,𝑉1 ,(𝑈ℎ ). ℎ∈𝐼(𝛼󸀠 ). .. Proof . Let Ψ(𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼󸀠 ) ) ∈ 𝐸𝜌,𝛼󸀠 ,𝑉0 ,𝑉1 ,(𝑈ℎ ) 󸀠 that we ℎ∈𝐼(𝛼 ) write in the form Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼󸀠 ) ). (81). ≤ 1,. and the lemma follows from the inequalities (79) and (81). Finally, the inequality (73) follows from (76) and (77).. =. ∑ 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼). 𝜓𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼󸀠 ) 𝑛. × Πℎ∈𝐼(𝛼)\𝐼(𝛼󸀠 ) 𝛿0,𝑙ℎ. 𝑛. 𝑙. 𝑈ℎ 𝑉0 0 𝑉1 1 Πℎ∈𝐼(𝛼) ℎ . 𝑛0 ! 𝑛1 ! 𝑙ℎ ! (85).

(16) 16. Abstract and Applied Analysis. By definition, we get that. 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) .. 󵄩󵄩 󵄩󵄩 󵄩󵄩𝜕𝑈 Ψ (𝑉0 , 𝑉1 , (𝑈ℎ ) 󵄩 󵄩󵄩 𝑗 󵄩󵄩𝜌,𝛼,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼) ℎ∈𝐼(𝛼󸀠 ) )󵄩 󵄨󵄨 󵄨󵄨 󵄨󵄨𝜓𝑛 ,𝑛 ,(𝑙 ) 󵄨 󵄨󵄨 0 1 ℎ ℎ∈𝐼(𝛼󸀠 ),ℎ ≠ 𝑗 ,𝑙𝑗 +1 Πℎ∈𝐼(𝛼)\𝐼(𝛼󸀠 ) 𝛿0,𝑙ℎ 󵄨󵄨󵄨 = ∑ exp (𝜎𝑟𝑏 (𝛼) 𝜌) 𝑛 ,𝑛 ,𝑙 ≥0,ℎ∈𝐼(𝛼) 0. 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 , (𝑈ℎ )ℎ≥0 , 𝑊) belong to 𝐺(𝜌,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ≥0 ,𝑊) . Then, the product 𝑏(𝑉0 , 𝑉1 , 𝑈0 , 𝑊)Ψ(𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊) belongs to 𝐺(𝜌,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ≥0 ,𝑊) and the following inequality:. 1 ℎ. 𝑛. ×. 𝑛. 𝑙. 𝑉00 𝑉11 Πℎ∈𝐼(𝛼) 𝑈ℎℎ (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼) 𝑙ℎ + 𝛼)!. . (86). We give upper bounds for this latter expression. 󵄩󵄩 󵄩 󵄩󵄩𝑏 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊) Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊)󵄩󵄩󵄩 󵄩 󵄩(𝜌,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ≥0 ,𝑊) ≤ |𝑏| (𝑉0 , 𝑉1 , 𝑈0 , 𝑊). 󵄩󵄩 󵄩󵄩 󵄩󵄩𝜕𝑈 Ψ (𝑉0 , 𝑉1 , (𝑈ℎ ) 󵄩 󵄩󵄩 𝑗 󵄩󵄩𝜌,𝛼,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼) ℎ∈𝐼(𝛼󸀠 ) )󵄩. 󵄩 󵄩 × 󵄩󵄩󵄩󵄩Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊)󵄩󵄩󵄩󵄩(𝜌,𝑉 ,𝑉 ,(𝑈 ) 0. 1. ℎ ℎ≥0 ,𝑊). (91). 󸀠. =. (. ∑. (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼󸀠 ),ℎ ≠ 𝑗 𝑙ℎ + 𝑙𝑗 + 1 + 𝛼 )! (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼󸀠 ) 𝑙ℎ + 𝛼)!. 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼󸀠 ). holds.. 1 × ) 𝑈𝑗 exp (𝜎𝜌 (𝑟𝑏 (𝛼) − 𝑟𝑏 (𝛼󸀠 ))). Proof. Let. 󵄨󵄨 󵄨󵄨 󵄨󵄨𝜓𝑛 ,𝑛 ,(𝑙 ) 󵄨 󵄨󵄨 0 1 ℎ ℎ∈𝐼(𝛼󸀠 ),ℎ ≠ 𝑗 ,𝑙𝑗 +1 󵄨󵄨󵄨 × exp (𝜎𝑟𝑏 (𝛼󸀠 ) 𝜌) 𝑛. ×. 𝑏 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊) = ∑ 𝑏𝛼 (𝑉0 , 𝑉1 , 𝑈0 ) 𝛼≥0. 𝑛. 𝑙. 𝑙 +1. 𝑉00 𝑉11 Πℎ∈𝐼(𝛼󸀠 ),ℎ ≠ 𝑗 𝑈ℎℎ 𝑈𝑗𝑗. 𝑊𝛼 , 𝛼!. Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊). .. (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼󸀠 ),ℎ ≠ 𝑗 𝑙ℎ + 𝑙𝑗 + 1 + 𝛼󸀠 )! (87). = ∑ Ψ𝛼 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼) ) 𝛼≥0. (92) 𝑊𝛼 . 𝛼!. Lemma 11. One has By definition, we get. (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼󸀠 ),ℎ ≠ 𝑗 𝑙ℎ + 𝑙𝑗 + 1 + 𝛼󸀠 )! (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼󸀠 ) 𝑙ℎ + 𝛼)! × ≤. 󵄩 󵄩󵄩 󵄩󵄩𝑏 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊) Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊)󵄩󵄩󵄩 󵄩(𝜌,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ≥0 ,𝑊) 󵄩. 1 exp (𝜎𝜌 (𝑟𝑏 (𝛼) − 𝑟𝑏 (𝛼󸀠 ))). (88). exp (−𝜎𝜌 ((𝛼 − 𝛼󸀠 ) /(𝛼 + 1)𝑏 )) 󸀠. 𝛼−𝛼 −1 Π𝑙=1 (𝛼 − 𝑙 + 1). .. 󵄩󵄩 󵄩󵄩 𝑏𝛼 (𝑉0 , 𝑉1 , 𝑈0 ) = ∑ 󵄩󵄩󵄩󵄩 ∑ 𝛼! 1 𝛼1 ! 󵄩󵄩𝛼1 +𝛼2 =𝛼 𝛼≥0 󵄩 ×. Proof. We notice that 𝛼. 1. 𝑟𝑏 (𝛼) − 𝑟𝑏 (𝛼󸀠 ) = ∑. 𝑛=𝛼󸀠 +1 (𝑛. + 1)𝑏. ≥. 𝛼 − 𝛼󸀠 (𝛼 + 1)𝑏. (89) Lemma 13. One has. and, with the help of (78), that for all integers 𝑎 ≥ 0, (𝑎 + 1 + 𝛼󸀠 )! (𝑎 + 𝛼)!. ≤. 1 󸀠. 𝛼−𝛼 −1 Π𝑙=1 (𝛼 − 𝑙 + 1). .. 󵄩 Ψ𝛼2 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼2 ) ) 󵄩󵄩󵄩 𝛼 󵄩󵄩 𝑊 . 󵄩󵄩 𝛼2 ! 󵄩󵄩 󵄩𝜌,𝛼,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼) (93). (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).. 󵄩󵄩 󵄩 󵄩󵄩𝑏𝛼1 (𝑉0 , 𝑉1 , 𝑈0 ) Ψ𝛼2 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼 ) )󵄩󵄩󵄩 󵄩 󵄩𝜌,𝛼,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼) 2 ≤. 𝛼2 ! 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑏 󵄨󵄨 (𝑉 , 𝑉 , 𝑈 ) 𝛼! 󵄨 𝛼1 󵄨 0 1 0 󵄩 󵄩 × 󵄩󵄩󵄩󵄩Ψ𝛼2 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼2 ) )󵄩󵄩󵄩󵄩 . 𝜌,𝛼2 ,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼2 ). (94).

(17) Abstract and Applied Analysis. 17 for all 𝛼 = 𝛼1 + 𝛼2 and all 𝑎 ∈ N which follows from (78). This yields the lemma. Using the fact that exp(𝜎𝑟𝑏 (𝛼)𝜌) ≥ exp(𝜎𝑟𝑏 (𝛼2 )𝜌) and gathering the inequalities (96) and (97) yield (94). Finally, using (93) with (94), one gets 󵄩󵄩 󵄩󵄩𝑏 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊). Proof. We can write 𝑏𝛼1 (𝑉0 , 𝑉1 , 𝑈0 ) =. ∑ 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼). 𝑏𝛼1 ,𝑛0 ,𝑛1 ,𝑙0 𝑛. × Πℎ∈𝐼(𝛼)\{0} 𝛿0,𝑙ℎ. 𝑙. 𝑛. 𝑈ℎ 𝑉0 0 𝑉1 1 Πℎ∈𝐼(𝛼) ℎ , 𝑛0 ! 𝑛1 ! 𝑙ℎ !. (95). Ψ𝛼2 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼2 ) ) =. ∑ 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼). 𝜓𝛼2 ,𝑛0 ,𝑛1 ,(𝑙ℎ )ℎ∈𝐼(𝛼 ) Πℎ∈𝐼(𝛼)\𝐼(𝛼2 ). 0. 1. ℎ ℎ≥0 ,𝑊). 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑏𝛼 󵄨󵄨 (𝑉0 , 𝑉1 , 𝑈0 ) ≤ ∑ ( ∑ 󵄨 1󵄨 𝛼1 ! 𝛼≥0 𝛼1 +𝛼2 =𝛼. 2. 𝑛. 󵄩 󵄩 × 󵄩󵄩󵄩󵄩Ψ𝛼2 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼2 ) )󵄩󵄩󵄩󵄩. 𝑙. 𝑛. 𝑈ℎ 𝑉0𝑉1 × 𝛿0,𝑙ℎ 0 1 Πℎ∈𝐼(𝛼) ℎ . 𝑛0 ! 𝑛1 ! 𝑙ℎ !. 𝜌,𝛼2 ,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼2 ). 𝛼. (100) from which the inequality (91) follows.. 󵄨 󵄨 ≤ 󵄨󵄨󵄨󵄨𝑏𝛼1 󵄨󵄨󵄨󵄨 (𝑉0 , 𝑉1 , 𝑈0 ). Proposition 15. (1) Let 𝑆, 𝑘 ≥ 0 be integers such that. ∑. 𝑛0 ,𝑛1 ,𝑙ℎ ≥0,ℎ∈𝐼(𝛼2 ). 𝑆 ≥ 𝑘 + 1 + max (𝑏 (𝑑1,𝑘 + 2) + 3, 𝑑 + 1 + 𝑏 (𝑑 + 𝑑1,𝑘 + 1)) . (101). 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨𝜓𝛼 ,𝑛 ,𝑛 ,(𝑙 ) 󵄨󵄨 2 0 1 ℎ ℎ∈𝐼(𝛼2 ) 󵄨󵄨󵄨 exp (𝜎𝑟𝑏 (𝛼) 𝜌). 𝑛. ×. 𝑛. 𝑙. 𝑉00 𝑉11 Πℎ∈𝐼(𝛼2 ) 𝑈ℎℎ (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼2 ) 𝑙ℎ + 𝛼)!. ).. Then, there exists a constant 𝐶8.1 > 0 (which is independent of 𝜌 > 1) such that 󵄩󵄩 −𝑆+𝑘 󵄩󵄩𝐵1,𝑘 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊) 𝜕𝑊 󵄩 󵄩 × DA Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊)󵄩󵄩󵄩󵄩(𝜌,𝑉 ,𝑉 ,(𝑈 ) ,𝑊) 0 1 ℎ ℎ≥0 (102) 𝑆−𝑘 ≤ 𝐶8.1 𝑊 󵄩 󵄩 × 󵄩󵄩󵄩󵄩Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊)󵄩󵄩󵄩󵄩(𝜌,𝑉 ,𝑉 ,(𝑈 ) 0. (96) Lemma 14. One has (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼2 ) 𝑙ℎ + 𝛼)! 𝛼2 ! 1 . 𝛼! (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼 ) 𝑙ℎ + 𝛼2 )!. (97). 2. Proof. We write 1 (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼2 ) 𝑙ℎ + 𝛼)! =. (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼2 ) 𝑙ℎ + 𝛼2 )!. ℎ ℎ≥0 ,𝑊). for all Ψ(𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊) ∈ 𝐺(𝜌,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ≥0 ,𝑊) . (2) Let 𝑆, 𝑘 ≥ 0 be integers such that (103). Then, there exists a constant 𝐶8.2 > 0 (which is independent of 𝜌 > 1) such that 󵄩󵄩 −𝑆+𝑘 󵄩󵄩𝐵2,𝑘 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊) 𝜕𝑊 󵄩 󵄩 × DB Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊)󵄩󵄩󵄩󵄩(𝜌,𝑉 ,𝑉 ,(𝑈 ) ,𝑊) 0 1 ℎ ℎ≥0 (104) 𝑆−𝑘 ≤ 𝐶8.2 𝑊 󵄩 󵄩 × 󵄩󵄩󵄩󵄩Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊)󵄩󵄩󵄩󵄩(𝜌,𝑉 ,𝑉 ,(𝑈 ). 1. 0. (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼2 ) 𝑙ℎ + 𝛼)! (𝑛0 + 𝑛1 + ∑ℎ∈𝐼(𝛼2 ) 𝑙ℎ + 𝛼2 )! (98). and we use the inequality (𝑎 + 𝛼2 )! 𝛼2 ! ≤ 𝛼! (𝑎 + 𝛼)!. 1. 𝑆 ≥ 𝑘 + 3 + 𝑏 (2 + 𝑑2,𝑘 ) .. 1. ≤. ). ×𝑊. By remembering (73) of Proposition 8, we deduce that 󵄩 󵄩󵄩 󵄩󵄩𝑏𝛼1 (𝑉0 , 𝑉1 , 𝑈0 ) Ψ𝛼2 (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ∈𝐼(𝛼 ) )󵄩󵄩󵄩 󵄩𝜌,𝛼,𝑉0 ,𝑉1 ,(𝑈ℎ )ℎ∈𝐼(𝛼) 󵄩 2. ×(. 󵄩 × Ψ (𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊)󵄩󵄩󵄩󵄩(𝜌,𝑉 ,𝑉 ,(𝑈 ). 1. ℎ ℎ≥0 ,𝑊). for all Ψ(𝑉0 , 𝑉1 , (𝑈ℎ )ℎ≥0 , 𝑊) ∈ 𝐺(𝜌,V0 ,𝑉1 ,(𝑈ℎ )ℎ≥0 ,𝑊) . Proof. (1) We show the first inequality (102). We expand 𝐵1,𝑘 (𝑉0 , 𝑉1 , 𝑈0 , 𝑊) = ∑ 𝐵1,𝑘,𝛼 (𝑉0 , 𝑉1 , 𝑈0 ) 𝛼≥0. (99) By definition, we have. 𝑊𝛼 . 𝛼!. (105).

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