Raising and lowering operators, factorization and differential/difference operators of hypergeometric type

TL;DR

This paper develops a unified framework for constructing raising and lowering operators for hypergeometric-type orthogonal polynomials, utilizing Rodrigues formulas, orthonormal functions, and the Infeld-Hull factorization method.

Contribution

It introduces a general method to derive mutually adjoint raising and lowering operators and the associated second-order hypergeometric operators for both continuous and discrete cases.

Findings

Constructed mutually adjoint raising and lowering operators.

Derived second-order hypergeometric operators via factorization.

Unified approach applicable to continuous and discrete orthogonal polynomials.

Abstract

Starting from Rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variable on uniform lattice. In order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight. Using the Infeld-Hull factorization method, we generate from the raising and lowering operators the second order self-adjoint differential/difference operator of hypergeometric type.