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A Whipple \(_7F_6\) Formula Revisited

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Abstract

A well-known formula of Whipple relates certain hypergeometric values \(_7F_6(1)\) and \(_4F_3(1)\). In this paper, we revisit this relation from the viewpoint of the underlying hypergeometric data HD, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple’s formula when the hypergeometric data HD are primitive and self-dual. If the data are also defined over \(\mathbb Q\), by the work of Katz, Beukers, Cohen, and Mellit, there are compatible families of \(\ell \)-adic representations of the absolute Galois group of \(\mathbb Q\) attached to HD. For specialized choices of HD, these Galois representations are shown to be decomposable and automorphic. As a consequence, the values of the corresponding hypergeometric character sums can be explicitly expressed in terms of Fourier coefficients of certain modular forms. We further relate the hypergeometric values \(_7F_6(1)\) in Whipple’s formula to the periods of these modular forms.

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Notes

  1. As an element in a global field, \(\lambda \) in different settings below is suitably interpreted: in (1), \(\lambda \) is viewed as a complex number via an embedding of the cyclotomic field in \(\mathbb C\), while in (2), \(\lambda \) is viewed as an element in the residue field of a nonarchimedean place of the cyclotomic field, when appropriate.

  2. The Legendre relation for L(k) is \(E(k)K(\sqrt{1-k^2})+K(k)E(\sqrt{1-k^2})-K(k)K(\sqrt{1-k^2})=\frac{\pi }{2}\), which is a pairing between \(H^0(L(k),\Omega _{V(1)})\) and \(H^1(L(k), {\mathcal {O}}_{V(1)})\).

  3. Here, e is set to be \(\frac{1-p}{2}\) instead of the target \(\frac{1}{2}\) so that the Gamma quotient on the right-hand side of (26) is a finite value.

  4. If we denote \( g(\tau )=f_{8.6.a.a}(\tau /2)\), then

    $$\begin{aligned} g(\tau \pm 1/2) =&i^{\pm } \left( \left( \frac{\eta (2\tau )^{3}}{\eta (\tau )\eta (4\tau )}\right) ^{12}-32\eta (2\tau )^{12}\left( \frac{\eta (4\tau )}{\eta (\tau )}\right) ^{4} \right) =i^{\pm }f_{16.6.a.a}\left( \frac{\tau }{2}\right) , \end{aligned}$$

    where \(f_{16.6.a.a}\) is the quadratic twist of \(f_{8.6.a.a}\) by \(\chi _{-1}\). It follows that

    $$\begin{aligned} _6F_5(HD_1(\frac{1}{2},\frac{1}{2}))= -4i \left( \int _{i/2}^{i\infty }f_{16.6.a.a}\left( \frac{\tau }{2}\right) (4\tau ^2+1) d\tau \right) . \end{aligned}$$
    (43)

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Acknowledgements

The research of Li is partially supported by Simons Foundation Grant # 355798. Long was supported in part by NSF DMS # 1602047 and the paper was written during her sabbatical leave in Fall 2020. The first two authors are grateful for their visits to the National Center for Theoretical Sciences in 2019. Moreover, the authors would like to thank Dr. Siu-Hung Ng, Wadim Zudilin, and the anonymous referees for helpful suggestions.

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  1. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Wen-Ching Winnie Li

  2. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Ling Long & Fang-Ting Tu

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Li, WC.W., Long, L. & Tu, FT. A Whipple \(_7F_6\) Formula Revisited. La Matematica 1, 480–530 (2022). https://doi.org/10.1007/s44007-021-00015-6

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