Clausen's theorem and hypergeometric functions over finite fields

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Abstract

We prove a general identity for a F23 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 F23 as the square of a F12. As another application, we evaluate an infinite family of F23(z) over Fq at z=1/8. This extends a result of Ono, who evaluated one of these F23(1/8) in 1998, using elliptic curves.

Keywords

Hypergeometric functions over finite fields
Clausen's theorem
Gegenbauer functions
Gauss sums
Jacobi sums

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