Web-seminar on Painlevé Equations and related topics

Next Seminar

20, December 2023 (Wed)   17:30～(JST)/9:30～(CET/UTC+1)

Changgui Zhang (Univ. of Lille)

On the connexion formula of a pantograph equation

Abstract : One calls a pantograph equation the following $q$-difference differential equation
$$y'(x)=ay(qx)+by(x)$$
where $0 < q < 1$ and $a b \not= 0$. Under the initial condition $y(0)=1$, one finds the power series solution
$$y_0(x)=1+\sum_{n \geq 1} \frac{(b+a)...(b+aq^{n-1})}{n!}x^n,$$
that is an entire function. Beside that, given any complex number $\nu$ such that $a\,q^\nu+b=0$, the series
$$g_\nu(x)= x^\nu\left(1+\sum_{n\ge" 1}\frac{(-\nu)(-\nu+1)...(-\nu+n-1)}{(1-q)(1-q^2)...(1-q^n)}\,q^{n(n+1)/2}\,(b\,x)^{-n}\right)$$
satisfies the above functional equation. The main question we will talk about in this seminar is the following: How to express $y_0$ using the system $\{g_{\nu}\mid a\,q^\nu+b=0\}$ ? Some related problems will also be considered during the talk.

Reference: C.Z. Analytical study of the pantograph equation using Jacobi theta functions, Journal of Approximation Theory, Volume 296, December 2023, 105974.

Painlevé Seminar Topics

Recently, we have encountered many interesting links between differential equations of Painlevé types and other fields of mathematics such as algebraic geometry, non-abelian Hodge theory and topological recursion and so on.
In this web seminar, we will organize research talks on Painlevé equations and related topics on web (zoom), basically about once every two or three weeks.
From September 2022, Regular time for seminar talk will be on Wednesday from 17:30--18:30 + discussion time in JST(Japan) or 10:30--11:30 in UTC+1. We are looking forward to your participation in the seminar. In order to get zoom access, please register through the form.

List of Talks

You can find the titles and abstracts of all seminars. ⇒ List of Talks

Registration

Online. Please register at the form, we will send you about web seminar address.

Organizers

Arata Komyo (Hyogo), Frank Loray (Rennes 1), Ryo Ohkawa (Osaka Metropolitan), Masa-Hiko Saito (Kobe Gakuin)

Scientific Committee (other than the Organizing Committee)

Takuro Mochizuki (RIMS, Kyoto), Claude Sabbah (Ecole polytechnique), Yosuke Ohyama (Tokushima U.)

Supported by

French ANR-16-CE40-0008 project "Foliage" (Frank Loray)

JSPS Grant-in-aid (A) 22H00094 (PI: Masa-Hiko Saito)

Osaka Central Advanced Mathematical Institute:

MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849.