Hypergeometric Functions Over Finite Fields

Abstract

ABSTRACT. 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.

Certain transformations for hypergeometric series in the p-adic setting

“…In [6], Greene introduced the notion of hypergeometric functions over finite fields or Gaussian hypergeometric series. He established these functions as analogues of classical hypergeometric functions.…”

Elliptic curves, eta-quotients and hypergeometric functions

“…Gaussian Hypergeometric Functions. In [Gre87], Greene defined Gaussian hypergeometric functions over arbitrary finite fields and showed that they have properties analogous to those of classical hypergeometric functions. We recall some definitions and notation from [Ono98] in the case of fields of prime order.…”

Gaussian Hypergeometric series and supercongruences

“…In [16] and [17], Greene defined general hypergeometric series over finite fields. His aim was to show that these functions satisfy properties analogous to classical hypergeometric series.…”

“…We extend all characters χ of F * p to F p by setting χ(0) := 0. Following [16] and [17], we give two definitions. The first definition is the finite field analogue of the …”

“…In this paper, we restrict our attention to the case A i = φ p for all i and B j = p for all j where φ p is the quadratic character and p is the trivial character mod p. We shall denote this value by n+1 F n (λ). By [16] and [17], p n n+1 F n (λ) ∈ Z. Before stating the main result, we recall that for i, n ∈ N, generalized Harmonic sums H (i) n are defined by…”