Translation operator with exceptional Laguerre polynomials

TL;DR

This paper extends the translation operator concept to exceptional Laguerre polynomials, analyzes related hyperbolic problems, and derives inequalities, contributing to the mathematical understanding of these special functions.

Contribution

It introduces a new translation operator for exceptional Laguerre polynomials and establishes its properties and applications, including Nikol'skii inequalities.

Findings

Derived a maximum principle for the associated hyperbolic Cauchy problem

Determined the norm of the translation operator

Established Nikol'skii inequalities for exceptional Laguerre polynomials

Abstract

We extend the notion of general translation operator to exceptional Laguerre polynomials. To this we investigate the associated singular hyperbolic Cauchy problem. We derive a maximum principle with respect to this Cauchy problem and applying it we determine the norm of the translation operator. As an application we give Nikol'skii inequalities with respect to exceptional Laguerre polynomials.