Summability of Formal Solutions for Some Generalized Moment Partial Differential Equations

Entire solutions of linear systems of moment differential equations and related asymptotic growth at infinity

“…In addition to this, the definition of moment differential operators can be extended not only to holomorphic functions near the origin, but also to summable functions. More precisely, given a strongly regular sequence M admitting a nonzero proximate order, and a function f defined on a finite sector S of opening larger than πω(M) and bisecting direction d ∈ R, which admits the formal power series f ∈ C[[z]] as its M-asymptotic expansion in S (observe from Corollary 4.12 [27] this function is unique under this property), then one can define ∂ m,z f as the unique function admitting ∂ m,z ( f ) as its M-asymptotic expansion in a finite sector of opening larger than πω(M) and bisecting direction d ∈ R. We refer to [15] for a more detailed description of this result.…”

“…Observe that the terms of the sum in (15) which correspond to the eigenvalue λ = 0 are such that E(λz) ≡ 1. This will have consequences regarding the asymptotic behavior of the solution at infinity.…”

“…The studies of moment differential equations were firstly developed by W. Balser and M. Yoshino in their former work [3], and have attracted the attention of many researchers in the last years due to the versatility of moment differentiation. Recently, several summability results of the formal solutions to moment partial differential equations have been obtained by S. Michalik [24] and by S. Michalik, M. Suwinska and the author [15], and also in the final section of the chapter [28], by J. Sanz. Also, some advances have been made on the study of moment integro-differential equations [17].…”

Summability of formal solutions for a family of generalized moment integro-differential equations

“…M p sequences (resp. regular M p sequences) of fixed order have been previously considered by the authors in [10,11] (resp. [10]) in the study of the estimates related to the formal solutions of moment partial differential equations, and the summability of such formal solutions.…”

“…So far, partial achievements have been accomplished regarding the summability of formal solutions of some families of generalized moment partial differential equations with variable coefficients. As a first step we cite [11], where the integral representation of functional moment derivatives allows to guarantee that the moment derivative of a summable function remains summable (see Theorem 3 and Corollary 1, [11]). This fact is applied to obtain summability results for the Cauchy problem…”