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Directions in Number Theory pp 125–159Cite as

Hypergeometric Series, Truncated Hypergeometric Series, and Gaussian Hypergeometric Functions

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Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 3))

Abstract

In this paper, we investigate the relationships among hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions through some families of “hypergeometric” algebraic varieties that are higher dimensional analogues of Legendre curves.

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Notes

  1. 1.

    The subscript q for a Gaussian hypergeometric function records the size of the corresponding finite field and should not be confused with the subscript for truncated hypergeometric series which records the location of truncation.

  2. 2.

    In a different language, our results correspond to the explicit descriptions of some mixed weight hypergeometric motives arising from exponential sums which are initiated by Katz [24], and are explicitly formulated and implemented by a group of mathematicians including Beukers, Cohen, Rodriguez-Villegas and others (from private communication with H. Cohen and F. Rodriguez-Villegas). Here we can use the explicit algebraic varieties to compute the Galois representations directly. A different algebraic model for the algebraic varieties is given in the following recent preprint [9].

  3. 3.

    When \(p \equiv 3\pmod 4\), \(_{2}F_{1}\left (\begin{array}{*{10}c} \eta _{2},&\eta _{2}\\ & \varepsilon \\ \end{array} \,; -1\right )_{p} = 0\).

  4. 4.

    Comparing both sides, when we pick \(u = -\zeta _{3}\) where ζ 3 be a primitive cubic root, we can derive that the claim of the theorem holds modulo p 3.

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Acknowledgements

We warmly thank the Banff International Research Station (BIRS) and Women in Numbers 3 BIRS 2014 (14w5009) workshop for the opportunity to initiate this collaboration. Thanks to the National Center for Theoretical Sciences in Taiwan for supporting the travel of Fang-Ting Tu to visit Ling Long. Long was supported by NSF DMS1303292. We are indebted to Wadim Zudilin for his insightful suggestions and sharing his ideas. We also thank Jerome W. Hoffman for his interest and valuable comments; Rupam Barman, Jesús Guillera, and Ravi Ramakrishna for helpful discussions. Further thanks to the referees for their careful readings and helpful comments. We used Magma and Sage for our computations related to this project and Sage Math Cloud to collaborate.

Author information

Authors and Affiliations

  1. Center for Communications Research, San Diego, CA, 92121, USA

    Alyson Deines

  2. Department of Mathematics and Computer Science, High Point University, High Point, NC, 27268, USA

    Jenny G. Fuselier

  3. Louisiana State University, Baton Rouge, LA, 70803, USA

    Ling Long

  4. Oregon State University, Corvallis, OR, 97331, USA

    Holly Swisher

  5. National Center for Theoretical Sciences, Hsinchu 300, Taiwan, R.O.C.

    Fang-Ting Tu

Authors
  1. Alyson Deines
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  2. Jenny G. Fuselier
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  3. Ling Long
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  4. Holly Swisher
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  5. Fang-Ting Tu
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Corresponding author

Correspondence to Ling Long .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, Oregon, USA

    Ellen E. Eischen

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

    Ling Long

  3. Department of Mathematics, Colorado State University, Fort Collins, Colorado, USA

    Rachel Pries

  4. Department of Mathematics, University of Colorado, Boulder, Colorado, USA

    Katherine E. Stange

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Deines, A., Fuselier, J.G., Long, L., Swisher, H., Tu, FT. (2016). Hypergeometric Series, Truncated Hypergeometric Series, and Gaussian Hypergeometric Functions. In: Eischen, E., Long, L., Pries, R., Stange, K. (eds) Directions in Number Theory. Association for Women in Mathematics Series, vol 3. Springer, Cham. https://doi.org/10.1007/978-3-319-30976-7_5

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