The hidden polynomial structure inside black hole ringing

Quasinormal modes govern how black holes "ring" after a merger, but their mathematical completeness and orthogonality remain poorly understood. By imposing quasinormal mode boundary conditions directly, the authors identify a class of polynomials that exactly tridiagonalize Teukolsky's radial equation for both Schwarzschild and Kerr geometries — and show these are Pollaczek-Jacobi polynomials with complex-valued parameters. A curious bonus: for Schwarzschild black holes, the three-term recurrence relation of these polynomials always peaks at the physical overtone index, linking an abstract algebraic property to observable mode labels.