**Financial support through the following projects:**

UID/MAT/00297/2019

UID/MAT/00297/2013

PEst-OE/MAT/UI0297/2014

PEst-OE/MAT/UI0297/2011

Analisys Seminar

Speaker: Davide Masoero, GFM/UL

Date: 2017, november, 8th

Time: 14 pm

Place: Room 1.9, building VII

Abstract:

In this talk I will briefly introduce the six Painleve equations, which are the only second order nonlinear ODEs whose solutions extend to the complex plane as meromorphic functions.

Afterwards, I will present a family of special solutions to the Painleve IV equation, whose singularities coincide with the roots of the generalised Hermite polynomials $H_{m,n}(z)$ (here m,n are two arbitrary positive integers).

I will show that roots of the generalised Hermite polynomials are encoded in the solution of a boundary value problem for a second order linear operator, which I will analyse to obtain the main results of the talk: 1)the computation of the number of real roots of $H_{m,n}(z)$ 2)the asymptotic distribution of roots of $H_{m,n}(z)$ for $|n+m|$ large. These generalises classical results about Hermite polynomials, see e.g. 'Higher transcendental functions, vol. II, chap. X'

This talk is based on a collaboration with P. Roffelsen.