  A family of orthogonal polynomials on the disk (which we call scattering
polynomials) serves to formulate a remarkable Fourier expansion of the
composition of a sequence of Poincar\'e disk automorphisms. Scattering
polynomials are tied to an exotic riemannian structure on the disk that is
hybrid between hyperbolic and euclidean geometries, and the expansion therefore
links this exotic structure to the usual hyperbolic one. The resulting identity
is intimately connected with the scattering of plane waves in piecewise
constant layered media. Indeed, a recently established combinatorial analysis
of scattering sequences provides a key ingredient of the proof. At the same
time, the polynomial obtained by truncation of the Fourier expansion elegantly
encodes the structure of the nonlinear measurement operator associated with the
finite time duration scattering experiment.
