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A scalar product for quasinormal mode solutions to Teukolsky's homogeneousradial equation is presented. Evaluation of this scalar product can beperformed either by direct integration, or by evaluation of a confluenthypergeometric functions. For direct integration, it is explicitly shown thatthe quasinormal modes' radial functions are regular on a family of physicallybounded complex paths. The related scalar product will be useful for betterunderstanding analytic solutions to Teukolsky's radial equation, particularlythe quasi-normal modes, their potential spatial completeness, and whether thequasi-normal mode overtone excitations may be estimated by spectraldecomposition rather than fitting. With that motivation, the scalar product isapplied to confluent Heun polynomials where it is used to derive their peculiarorthogonality and eigenvalue properties. A potentially new relationship isderived between the confluent Heun polynomials' scalar products andeigenvalues. Using these results, it is shown for the first time thatTeukolsky's radial equation (and perhaps similar confluent Heun equations) are,in principle, exactly tridiagonalizable. To this end, "canonical" confluentHeun polynomials are conjectured.

A radial scalar product for Kerr quasinormal modes