Searcharxiv⌕ Search

arXiv subjects

Sławomir Michalik

Publications and source records attributed to Sławomir Michalik.

At least 19 recordsLinked to original sources
arXiv2026

Sequences of generalized moments and their applications to some moment partial differential equations

We introduce sequences of generalized moments that are a generalization of sequences preserving summability and sequences of moments from Balser's theory of moment summability. We construct extensions of Laplace and Borel integral operators for these new sequences. We find integral representations of moment derivatives for generalized moments. We show the characterization of summable solutions of moment-PDEs with generalized moments. We describe singular directions for summable solutions of moment-PDEs with generalized moments and we express jumps across these singular directions in terms of hyperfunctions and in terms of kernel functions associated with sequences preserving summability, which are connected with given sequences of generalized moments.

math.AP↗
arXiv2026

Summability of formal solutions of some singular perturbations problems in differential and moment differential equations

In this paper we study the summability of solutions of some general forms of singularly perturbed linear ordinary differential and moment differential equations. We conclude that under some assumptions solutions of these equations are summable. The type of this summability depends on the specific equation. We also show the connection between some singularly perturbed moment ordinary differential equations and some linear moment partial differential equations. We apply this connection to describe summable and multisummable formal solutions of these singularly perturbed moment ordinary differential equations. Main techniques used to show these conclusions are based on Borel transforms, properties of solutions of moment partial differential equations and on the Cauchy integral formula together with integral representations of solutions of such equations.

math.CA↗
arXiv2025

On conditions determining formal automorphisms of integro-differential operators

Results providing conditions on a family of integro-differential operators to determine a formal automorphism are established. Equivalently, the problem can be read in terms of existence and uniqueness of formal solutions of Cauchy problems in different settings. The main achievements provide a three-statement result not only in the framework of general formal power series, but also on subspaces appearing in applications such as Gevrey settings and moment differential operators.

math.CV↗
arXiv2025

On sequences preserving summability

In this paper, we study sequences of positive numbers preserving summability. In particular, the open set property for such a family of sequences is shown. Several classes of sequences preserving summability, including polynomials, sums of powers, and solutions of some difference equations, are introduced.

math.CA↗
arXiv2024

On integral representations of $q$-difference operators and their applications

Integral representations of two $q$-difference operators are provided in terms of special functions arising in the theory of asymptotic solutions to $q$-difference equations in the complex domain. Both representations are unified through the so-called $(p,q)$-differential operator, for which a kernel-like function is provided, generating the sequence of $(p,q)$-factorials.

math.CV↗
arXiv2023

On the summability and convergence of formal solutions of linear $q$-difference-differential equations with constant coefficients

We consider the Cauchy problem for homogeneous linear $q$-difference-differential equations with constant coefficients. We characterise convergent, $k$-summable and multisummable formal power series solutions in terms of analytic continuation properties and growth estimates of the Cauchy data. We also introduce and characterise sequences preserving summability, which make a very useful tool, especially in the context of moment differential equations.

math.AP↗
arXiv2023

On polyharmonic polynomials

We study the orthogonal projection of homogeneous polynomials onto the space of homogeneous polyharmonic polynomials. To do this we derive the decomposition of homogeneous polynomials in terms of the Kelvin transform of derivatives of the fundamental solution $|x|^{2-n}$ or $\log |x|$. We consider also the vector bases of the space of homogeneous polyharmonic polynomials and study the problem of convergence of orthogonal series.

math.CA↗
arXiv2023

On sequences preserving q-Gevrey asymptotic expansions

The modification of the coefficients of formal power series is analyzed in order that such variation preserves q-Gevrey asymptotic properties, in particular q-Gevrey asymptotic expansions. A characterization of such sequences is determined, providing a handy tool in practice. The sequence of q-factorials is proved to preserve q-Gevrey asymptotic expansions.

math.CV↗
arXiv2022

Multisummability of formal solutions for a family of generalized singularly perturbed moment differential equations

The notion of moment differentiation is extended to the set of generalized multisums of formal power series via an appropriate integral representation and accurate estimates of the moment derivatives. The main result is applied to characterize generalized multisummability of the formal solution to a family of singularly perturbed moment differential equations in the complex domain, broadening widely the range of singularly perturbed functional equations to be considered in practice, such as singularly perturbed differential equations and singularly perturbed fractional differential equations.

math.CA↗
arXiv2020

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

Generalized summability results are obtained regarding formal solutions of certain families of linear moment integro-differential equations with time variable coefficients. The main result leans on the knowledge of the behavior of the moment derivatives of the elements involved in the problem. A refinement of the main result is also provided giving rise to more accurate results which remain valid in wide families of problems of high interest in practice, such as fractional integro-differential equations.

math.AP↗
arXiv2019

Estimates of formal solutions for some generalized moment partial differential equations

Using increasing sequences of real numbers, we generalize the idea of formal moment differentiation first introduced by W. Balser and M. Yoshino. Slight departure from the concept of Gevrey sequences enables us to include a wide variety of operators in our study. Basing our approach on tools such as the Newton polygon and divergent formal norms, we obtain estimates for formal solutions of certain families of generalized linear moment partial differential equations with constant and time variable coefficients.

math.AP↗
arXiv2019

Gevrey estimates for certain moment partial differential equations

We consider the Cauchy problem for inhomogeneous linear moment differential equations with holomorphic time dependent coefficients. Using such tools as the formal norms, theory of majorants and the properties of the Newton polygon, we obtain the Gevrey estimate for the formal solution of the equation.

math.AP↗
arXiv2018

The Stokes phenomenon for some moment partial differential equations

We study the Stokes phenomenon for the solutions of general homogeneous linear moment partial differential equations with constant coefficients in two complex variables under condition that the Cauchy data are holomorphic on the complex plane but finitely many singular or branching points with the appropriate growth condition at the infinity. The main tools are the theory of summability and multisummability, and the theory of hyperfunctions. Using them we describe Stokes lines, anti-Stokes lines, jumps across Stokes lines, and a maximal family of solutions.

math.AP↗
arXiv2018

Hyperasymptotic solutions for certain partial differential equations

We present the hyperasymptotic expansions for a certain group of solutions of the heat equation. We extend this result to a more general case of linear PDEs with constant coefficients. The generalisation is based on the method of Borel summability, which allows us to find integral representations of solutions for such PDEs.

math.AP↗
arXiv2018

Spherical polyharmonics and Poisson kernels for polyharmonic functions

We introduce and develop the notion of spherical polyharmonics, which are a natural generalisation of spherical harmonics. In particular we study the theory of zonal polyharmonics, which allows us, analogously to zonal harmonics, to construct Poisson kernels for polyharmonic functions on the union of rotated balls. We find the representation of Poisson kernels and zonal polyharmonics in terms of the Gegenbauer polynomials. We show the connection between the classical Poisson kernel for harmonic functions on the ball, Poisson kernels for polyharmonic functions on the union of rotated balls, and the Cauchy-Hua kernel for holomorphic functions on the Lie ball.

math.AP↗
arXiv2017

A Dirichlet type problem for complex polyharmonic functions

We extend holomorphically polyharmonic functions on a real ball to a complex set being the union of rotated balls. We solve a Dirichlet type problem for complex polyharmonic functions with the boundary condition given on the union of rotated spheres.

math.AP↗