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

For a∈Zpa\in\mathbb{Z}_p, let l∈{0,…,p−1}l\in\{0,\ldots,p-1\} be the unique integer such that a+l≡0(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

Zp⟨t,t−1,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(t−1)f(t^{-1}).

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

Fa,…,aDw(t)=((−1)d+1t)lFa,…,aDw(t−1).\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 t−1t^{-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.

References

Primary source

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

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