Semi-classical Laguerre polynomials and a third order discrete integrable equation

TL;DR

This paper derives a third order discrete integrable equation from semi-classical Laguerre polynomials using a semi-discrete Lax pair, advancing understanding of semi-classical orthogonal polynomials and their integrable structures.

Contribution

It introduces a new third order difference equation associated with semi-classical Laguerre polynomials and constructs its Lax pair, linking orthogonal polynomials to integrable systems.

Findings

Derived a third order discrete integrable equation

Constructed a semi-discrete Lax pair for the equation

Connected semi-classical Laguerre polynomials to integrable systems

Abstract

A semi-discrete Lax pair formed from the differential system and recurrence relation for semi-classical orthogonal polynomials, leads to a discrete integrable equation for a specific semi-classical orthogonal polynomial weight. The main example we use is a semi-classical Laguerre weight to derive a third order difference equation with a corresponding Lax pair.