A quadratic hypergeometric ₂𝐹₁ transformation over finite fields

Evans, Ron; Greene, John

doi:10.1090/proc/13303

A quadratic hypergeometric ${}_2\hspace {-1pt}F_1$ transformation over finite fields
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Proc. Amer. Math. Soc. 145 (2017), 1071-1076 Request permission

Abstract:

In 1984, the second author conjectured a quadratic transformation formula which relates two hypergeometric ${}_2\hspace {-1pt}F_1$ functions over a finite field $\mathbb {F}_q$. We prove this conjecture in Theorem 2. The proof depends on a new linear transformation formula for pseudo-hypergeometric functions over $\mathbb {F}_q$. Theorem 2 is then applied to give an elegant new transformation formula (Theorem 3) for ${}_2\hspace {-1pt}F_1$ functions over finite fields.

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