Systems of orthogonal polynomials defined by hypergeometric type equations with application to quantum mechanics

TL;DR

This paper provides a unified framework for systems of orthogonal polynomials derived from hypergeometric type equations, linking them to quantum mechanics applications like Schrödinger equations and bound-state eigenfunctions.

Contribution

It explicitly characterizes all such orthogonal polynomial systems, their special functions, and raising/lowering operators, connecting mathematical structures to quantum physics models.

Findings

Unified explicit description of orthogonal polynomial systems

Connection to Schrödinger-type equations and bound states

Development of raising/lowering operators for these systems

Abstract

A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated special functions and the corresponding raising/lowering operators. The considered equations are directly related to some Schrodinger type equations (Poschl-Teller, Scarf, Morse, etc), and the defined special functions are related to the corresponding bound-state eigenfunctions.