Analytical study of the pantograph equation using Jacobi theta functions

Zhang, Changgui

Type de document :

Compte-rendu et recension critique d'ouvrage

DOI :

10.1016/j.jat.2023.105974

Titre :

Analytical study of the pantograph equation using Jacobi theta functions

Auteur(s) :

Zhang, Changgui [Auteur]

Laboratoire Paul Painlevé - UMR 8524 [LPP]

Titre de la revue :

Journal of Approximation Theory

Éditeur :

Elsevier

Date de publication :

2023

ISSN :

0021-9045

Mot(s)-clé(s) en anglais :

Pantograph equation
q-difference equation
Connection problem
Jacobi θ-function

Discipline(s) HAL :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]

Résumé en anglais : [en]

The aim of this paper is to use the analytic theory of linear q-difference equations for the study of the functional-differential equation y ′ (x) = ay(q x) + by(x), where a and b are two non-zero real or complex numbers. ...
Lire la suite >The aim of this paper is to use the analytic theory of linear q-difference equations for the study of the functional-differential equation y ′ (x) = ay(q x) + by(x), where a and b are two non-zero real or complex numbers. When 0 < q < 1 and y(0) = 1, the associated Cauchy problem admits a unique power series solution, ∑ n≥0 (-a/b; q) n/n! (bx) n , that converges in the whole complex x-plane. The principal result obtained in the paper explains how to express this entire function solution into a linear combination of solutions at infinity with the help of integral representations involving Jacobi theta functions. As a byproduct, this connection formula between zero and infinity allows one to rediscover the classic theorem of Kato and McLeod on the asymptotic behavior of the solutions over the real axis.Lire moins >

Langue :

Anglais

Vulgarisation :

Non

Projet ANR :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :