Research on the recurrence relations for the spin-weighted spheroidal harmonics

TL;DR

This paper investigates recurrence relations for spin-weighted spheroidal harmonics using supersymmetric quantum mechanics, establishing new relations that connect different spin-weights and enabling derivation from spheroidal to spherical harmonics.

Contribution

It introduces the first recurrence relations for SWSHs based on shape invariance, linking them to spherical harmonics and advancing theoretical and astrophysical applications.

Findings

Derived recurrence relations for SWSHs with the same m but different s

Connected SWSHs to spherical harmonics through these relations

Demonstrated the importance of these relations in astrophysical contexts

Abstract

In this paper we study the recurrence relations in the spin-weighted spheroidal harmonics (SWSHs) through super-symmetric quantum mechanics. We use the shape invariance property to solve the spin-weighted spheroidal wave equations. The result shows the relation among SWSHs with a special condition of the same parameter magnetic quantum number m but different spin-weight s. The conclusions can be reduced to the famous recurrence relations of spin-weighted spherical harmonics. These contents are the first investigation of this kind recurrence relation concerning SWSHs and make it possible to derive SWSHs from the spheroidal harmonics, so they are very important both in theoretical background and in the astrophysical applications. Keywords: spin-weighted spheroidal harmonics, recurrence relation, super-symmetric quantum mechanics, shape-invariance