Wang's transformation conjecture for Dwork's p-adic hypergeometric function

For aZpa\in\mathbb{Z}_p, let l{0,,p1}l\in\{0,\ldots,p-1\} be the unique integer such that a+l0(modp)a+l\equiv0\pmod p. Let Fa,,aDw(t)\mathscr{F}^{\rm Dw}_{a,\ldots,a}(t) be Dwork's pp-adic hypergeometric function, viewed in

Zpt,t1,ha,,a(t)1.\mathbb{Z}_p\langle t,t^{-1},h_{a,\ldots,a}(t)^{-1}\rangle.

The involution ι\iota sends f(t)f(t) to f(t1)f(t^{-1}).

Wang's transformation conjecture. If pp is an odd prime, then

Fa,,aDw(t)=((1)d+1t)lFa,,aDw(t1).\mathscr{F}^{\rm Dw}_{a,\ldots,a}(t)=((-1)^{d+1}t)^l\mathscr{F}^{\rm Dw}_{a,\ldots,a}(t^{-1}).

If p=2p=2, the same identity holds up to sign.

This conjecture proposes a transformation formula relating Dwork's pp-adic hypergeometric function at tt and t1t^{-1}, analogous to classical hypergeometric transformation formulas. The cited work of Wang formulates the conjecture, while its resolution is not established in the supplied source.

Sources & referencesView supporting material

Primary source

Yusuke Nemoto, “Transformation formula of Dwork's p-adic hypergeometric function”, arXiv:2505.24215 (2025).

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