  In this paper we study sharp estimates for the Schr\"odinger operator via the
framework of orthogonal polynomials. We use spherical harmonics and Gegenbauer
polynomials to prove a new weighted inequality for the Schr\"odinger equation
that is maximized by radial functions. We use Hermite and Laguerre polynomial
expansions to produce sharp Strichartz estimates for even exponents. In
particular, for radial initial data in dimension 2, we establish an interesting
connection of the Strichartz norm with a combinatorial problem about words with
four letters.
