Ultradiscrete limit of Bessel function type solutions of the Painlev\'{e} III equation

TL;DR

This paper constructs an ultradiscrete analog of the Bessel function via ultradiscrete limits of a q-difference Bessel function and explores its connection to special solutions of the ultradiscrete Painlevé III equation.

Contribution

It introduces a new ultradiscrete Bessel function and links it to special solutions of ultradiscrete Painlevé III with determinantal structure.

Findings

Ultradiscrete Bessel function constructed from q-difference analog.

Established relationship between ultradiscrete and discrete Painlevé III solutions.

Connected special solutions with determinantal structure across ultradiscrete and discrete forms.

Abstract

An ultradiscrete analog of the Bessel function is constructed by taking the ultradiscrete limit for a $q$-difference analog of the Bessel function. Then, a direct relationship between a class of special solutions for the ultradiscrete Painlev\'{e} III equation and those of the discrete Painlev\'{e} III equation which have a determinantal structure is established.