Sum rules for products of light scattering functions

Basis vectors of a spherical system are shown to be linear combinations of functions commonly used in light scattering. These expressions are used to obtain an expansion for the unit dyadic as a linear combination of products of these functions, and this expansion is used to motivate the use of matrix orthonormality to obtain a complete set of sum rules for products of scalar light scattering functions. As an example demonstrating the utility of these expressions, sum rules are obtained for dyadic, component, and inner products of vector spherical harmonics used in the theory of light scattering.