Egocentric physics: Summing up Mie

We consider sum rules for the electric type (TM) multipole coefficients in the Mie theory of the scattering coefficients for electromagnetic waves incident upon spherical particles. These sum rules are derived from infinite product representations for the scattering coefficients and involve an analytically-determined multiplying factor in addition to the resonant eigenstate values. The product expansions converge rapidly to the scattering coefficients with increasing numbers of resonant state values only if the analytic multiplying factor is included in expansions, and the use of these sum rules further accelerates the convergence of scattering coefficient expansions. We present analytic asymptotic estimates for the resonant state eigenvalues in the dipole, quadrupole and hexapole cases, give the corresponding sum rules, and numerically illustrate their convergence.