  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. The
first author recently gave such a proof of his identities when q = 1 (mod 4),
and this paper provides such a proof for the remaining case q = 3 (mod 4). Our
proofs are valid for all characteristics p > 2. Along the way we prove some
elegant new character sum identities.
