Factorization method and general second order linear difference equation

TL;DR

This paper explores a factorization approach for second order linear difference equations, demonstrating how certain classes, including those related to classical orthogonal polynomials, can be factorized using adjoint first order operators.

Contribution

It introduces a factorization method for second order difference operators in Hilbert spaces, covering hypergeometric type equations related to discrete orthogonal polynomials.

Findings

Certain second order difference operators can be factorized into first order operators.

The factorization applies to hypergeometric type equations.

This approach links difference equations to orthogonal polynomial theory.

Abstract

This paper addresses an investigation on a factorization method for difference equations. It is proved that some classes of second order linear difference operators, acting in Hilbert spaces, can be factorized using a pair of mutually adjoint first order difference operators. These classes encompass equations of hypergeometic type describing classical orthogonal polynomials of a discrete variable.