Hypergeometric Functions for Function Fields

Fractional hypergeometric functions for function fields and weak transcendence in positive characteristic

Saturday, Sep 23 On some results and problems involving Hardy’s function Z(t) Aleksandar Ivić Serbian Academy of Arts and Sciences, Belgrade Hardy’s classical function Z(t) is defined, for real t, as Z(t) := ζ( 1 2 + it)χ( 1 2 + it)−1/2, ζ(s) = χ(s)ζ(1− s), This talk will cover some problems involving Hardy’s function, including moments at points where |Z(t)| is maximal, large values of Hardy’s...

Special Values of Hypergeometric Functions over Finite Fields

For an odd prime p, define Hp(z) = ∑ u,v(mod p) ( uv(1−u)(1−v)(1−uvz) p ) , where z is an integer (mod p) and the summands are Legendre symbols. The function Hp(z) was explicitly evaluated for z = 1 by Evans (1981) and for z = −1 by Greene and Stanton (1986). Koike (1992) determined Hp(1/4)(mod p), and Ono (1998) evaluated Hp(z) for z = 1/4,−1/8, and 1/64. This paper evaluates Hp(z) for infinit...

Clausen's theorem and hypergeometric functions over finite fields

We prove a general identity for a 3F2 hypergeometric function over a finite field Fq, where q is a power of an odd prime. A special case of this identity was proved by Greene and Stanton in 1986. As an application, we prove a finite field analogue of Clausen’s Theorem expressing a 3F2 as the square of a 2F1. As another application, we evaluate an infinite family of 3F2(z) over Fq at z = −1/8. T...

Transformations of well-poised hypergeometric functions over finite fields

We define a hypergeometric function over finite fields which is an analogue of the classical generalized hypergeometric series. We prove that this function satisfies many transformation and summation formulas. Some of these results are analogous to those given by Dixon, Kummer and Whipple for the well-poised classical series. We also discuss this function’s relationship to other finite field an...

Hypergeometric Functions