Hermite Character Sums

This evaluation proves a conjecture in [9, p. 370] and solves the problem of finding explicitly the number of rational points (mod/?) on the surface z = (x + l)(y + ΐ)(x + y), a problem some algebraic geometers had worked on without success. Character sum analogues of the important formulas for orthogonal polynomials are potentially as useful as those for hypergeometric series, so a systematic study should be made. Indeed, many character sums studied in the literature are analogues of special functions, e.g., the generalized Kloosterman sum (see (2.5), Theorem 2.6, and, say, [10], [21A, p. 253]). In this paper, the focus is on analogues of Hermite polynomials, namely Hermite character sums HN(x) defined in (2.1). Each of the theorems in §4 is an analogue over finite fields of a classical formula stated just above it. The classical formulas are stated without conditions of validity; such conditions are often unrelated to the unpredictable conditions of validity for the finite field formulas. It is not always possible to give proofs of the finite field formulas which parallel classical proofs. This is because no satisfactory analogues of limits, first derivatives, logarithms, and three term recurrence relations are known. It would be of great importance to find a unified approach which simultaneously explains formulas for orthogonal polynomials and the analogues over finite fields. Perhaps this will be accomplished by connecting the polynomials with Lie groups having counterparts over finite fields.