# A new sum rule for Clebsch-Gordan coefficients using generalized characters of irreducible representations of the rotation group

## TL;DR

This paper introduces a novel sum rule for Clebsch-Gordan coefficients derived from generalized characters of rotation group representations, utilizing integrals with Gegenbauer polynomials, potentially enabling new mathematical relations.

## Contribution

The paper presents a new sum rule for Clebsch-Gordan coefficients based on generalized characters and integral identities involving Gegenbauer polynomials, expanding the mathematical tools available.

## Key findings

- Derived a new sum rule for Clebsch-Gordan coefficients
- Connected the sum rule to integrals of Gegenbauer polynomials
- Suggested applicability to other polynomial integral relations

## Abstract

We present a new sum rule for Clebsch-Gordan coefficients using generalized characters of irreducible representations of the rotation group. The identity is obtained from an integral involving Gegenbauer ultraspherical polynomials. A similar procedure can be applied for other types of integrals of such polynomials, and may therefore lead to the derivation of further new relations.