Super Catalan Numbers and Fourier Summation over Finite Fields

We study polynomial summation over unit circles over finite fields of odd characteristic, obtaining a purely algebraic integration theory without recourse to infinite procedures. There are nonetheless strong parallels to classical integration theory over a circle, and we show that the super Catalan numbers and closely related rational numbers lie at the heart of both theories. This gives a uniform analytic meaning to these up to now somewhat mysterious numbers. Our derivation utilises the three-fold symmetry of chromogeometry between Euclidean and relativistic geometries, and we find that the Fourier summation formulas we derive in these two different settings are closely connected.