Fields of Definition of Finite Hypergeometric Functions

Abstract

Finite hypergeometric functions are functions of a finite field F q to C. They arise as Fourier expansions of certain twisted exponential sums and were introduced independently by John Greene and Nick Katz in the 1980's. They have many properties in common with their analytic counterparts, the hypergeometric functions. One restriction in the definition of finite hypergeometric functions is that the hypergeometric parameters must be rational numbers whose denominators divide q − 1. In this note we use the symmetry in the hypergeometric parameters and an extension of the exponential sums to circumvent this problem as much as posssible.

A Whipple $$_7F_6$$ Formula Revisited

“…In recent years, rapid progress has been made in hypergeometric character sums under the framework of hypergeometric motives. It gives a unifying perspective to many hypergeometric structures and identities, see [8,12,18,19,[27][28][29]32,38,42,49]. Classical hypergeometric formulas such as Clausen's formula have a finite field analogue by Evans and Greene [22].…”

“…) and is isomorphic to the level 4 index-6 subgroup of SL 2 (Z) labeled by 4C 0 in [16]. The group 4C 0 is a supergroup of 0 (8). By the above dimension formula, dim S 6 ( ( 12 , 1…”

“…= 16 • 2πiτ 2 • f 8.6.a.a (τ/2)dτ , where f 8.6.a.a (τ/2) = η 12 (τ ) + 32η 4 (τ )η(4τ ) 8 = − η 12 (τ ) − 2 η(2τ ) 24η(τ ) 4 η(4τ )8 . Using (41), we get the first claim of Theorem 6.Next we demonstrate the appearance of g c, f = f 8.4.a.a (τ ) = η(2τ ) 4 η(4τ )4 in the second claim of Theorem 6 by showing where the notation ∼ e means the two differentials differ by an exact differential form.Finally we consider the path |t 2 | = 1.…”

Some Numeric Hypergeometric Supercongruences

“…To relate the finite hypergeometric sum H p (α, β; λ) defined in (5) to the truncated hypergeometric functions, one uses the Gross-Koblitz formula [19] which says for integer k…”

“…When β = {1, • • • , 1} the details of a proof is already given in Section 4 of [30]. Another reference is Beukers' paper [5].…”