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Properties of natural polynomials for Schwarzschild and Kerr black holes

Michelle Foucoin and Lionel London

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Phys. Rev. D 114, 084039 – Published 13 October, 2026

DOI: https://doi.org/10.1103/wnb8-vndc

Abstract

The quasinormal modes of black holes play various important roles in gravitational wave theory, signal modeling, and data analysis; however, there remain open questions about their mathematical properties. Aspects of classical polynomial theory have been proposed as a framework to investigate quasinormal mode orthogonality and completeness. We have recently presented a class of polynomials that are “natural” to quasinormal modes in that they are restricted by the quasinormal mode boundary conditions, and exactly tridiagonalize Teukolsky’s radial equation. In turn, these polynomials may be useful for (i) better understanding the vector space properties of quasinormal mode solutions (ii) serving as a basis for spectral methods to solve the Teukolsky and confluent Heun equations. Here, we provide an overview of these polynomials’ basic analytic properties: their three-term recurrence relation, ladder operators, and governing differential equation. We demonstrate that the natural polynomials for Schwarzschild and Kerr black holes are Pollaczek-Jacobi polynomials with complex valued parameters. We verify that, despite being constrained to the real line, key results for Pollaczek-Jacobi polynomials hold for black hole polynomials. Along the way, we observe a novel property that is particular to Schwarzschild: the polynomials’ three-term recurrence relation always peaks at the physical overtone index. This work supports the broader application of these polynomials to spectral methods for solving the Teukolsky equation, and other instances of the confluent Heun equation.

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Phys. Rev. D 114, 084039– Published 13 October, 2026
  • Received 15 May 2026
  • Accepted 9 September 2026
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