John Greene

Hypergeometric functions over finite fields

In this paper the analogy between the character sum expansion of a complex-valued function over GF(q) and the power series expansion of an analytic function is exploited in order to develop an analogue for hypergeometric series over finite fields. It is shown that such functions satisfy many summation and transformation formulas analogous to their classical counterparts.

A character sum evaluation and Gaussian hypergeometric series

Abstract Evans has conjectured the value of a certain character sum. The conjecture is confirmed using properties of Gaussian hypergeometric series which are well known for hypergeometric series. Several related questions are discussed.

The triplication formula for Gauss sums

A new proof of the triplication formula for Gauss sums is given. It mimics an old proof of the analogous result for gamma functions. The techniques are formal and rely upon the character properties of fields. A new character sum evaluation is given.

Periodic solutions to some difference equations over the integers

In this paper, we investigate periodic integer solutions {a n } to where r is a rational number. We show that solutions can only exist, if − 1 ≤ r ≤ 1/2 and we give several infinite families of rs, for which the above recurrence has periodic solutions in the integers.

Polynomials with nonnegative coefficients whose zeros have modulus one

Define

Some mixed character sum identities of Katz II

p(z) = \prod _{j = 0}^{n - 1} (z - e^{i(\theta + \alpha j)} )

TRACES OF MATRIX PRODUCTS

for

Greedy Matching in Young's Lattice

\alpha > 0

Linearized polynomials and permutation polynomials of finite fields.

and

Evaluations of hypergeometric functions over finite fields

\theta \geq 0