From Сlausen to Сarlitz: low-dimensional spin groups and identities among character sums

We relate the classical formulas of Clausen and Schläfli for the squares of hypergeometric and Bessel functions respectively, and a 1969 formula of Carlitz for the square of a very particular Kloosterman sum, to the “accident” that for $3\le n\le6$, the “spin” double cover of $SO(n)$ is itself a classical group. We exploit this accident to obtain identities among character sums over finite fields, some but not all of which are finite field analogues of known identities among classical functions.