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 $q \equiv 3 \pmod 4$, and in this paper we give a proof for the remaining case $q \equiv 1 \pmod 4$. Moreover, we show that the identities are valid for all characteristics $p >2$.