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)
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.