Exact and simple formulas for the linearization coefficients of products of orthogonal polynomials and physical application

TL;DR

This paper derives simple, exact formulas for the linearization coefficients of orthogonal polynomial products, facilitating numerical calculations and revealing a physical phenomenon involving spectrum mixing due to nonlinear coupling.

Contribution

It provides general, compact formulas for linearization coefficients of orthogonal polynomials based on three-term recursion coefficients, applicable to a wide class of polynomials.

Findings

Formulas are more suitable for numerical calculations than previous ones.

Applicable to both conventional and modified linearization coefficients.

Demonstrates a physical phenomenon where nonlinear coupling causes spectrum mixing.

Abstract

We obtain exact, simple and very compact expressions for the linearization coefficients of the products of orthogonal polynomials; both the conventional Clebsch-Gordan-type and the modified version. The expressions are general depending only on the coefficients of the three-term recursion relation of the linearizing polynomials. These are more appropriate and useful for doing numerical calculations when compared to other exact formulas found in the mathematics literature, some of which apply only to special class of polynomials while others may involve the evaluation of intractable integrals. As an application in physics, we present a remarkable phenomenon where nonlinear coupling in a physical system with pure continuous spectrum generates a mixed spectrum of continuous and discrete energies.