Orthogonal polynomials for the weight $x^{\nu} \exp(-x - t/x)$

TL;DR

This paper studies orthogonal polynomials associated with a specific weight function involving exponential and inverse terms, deriving their differential equations, recurrence relations, explicit formulas, and generating functions.

Contribution

It provides new explicit representations and formulas for orthogonal polynomials with the weight $x^{ u} e^{-x - t/x}$, expanding understanding of their properties.

Findings

Derived differential-difference equations for the polynomials.

Established recurrence relations and Rodrigues-type formulas.

Obtained explicit representations and generating functions.

Abstract

Orthogonal polynomials for the weight $x^{\nu} \exp(-x - t/x),\ x, t > 0, \nu \in \mathbb{R}$ are investigated. Differential-difference equations, recurrence relations, explicit representations, generating functions and Rodrigues-type formula are obtained.