Some mixed character sum identities of Katz

A conjecture connected with quantum physics led N. Katz to discover some amazing mixed character sum identities over a field of q elements, where q is a power of a prime p>3. His proof required deep algebro-geometric techniques, and he expressed interest in finding a more straightforward direct proof. Such a proof has been given by Evans and Greene in the case q≡3(mod4), and in this paper we give a proof for the remaining case q≡1(mod4). Moreover, we show that the identities are valid for all characteristics p>2.