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 Uspekhi Mat. Nauk, 2011, Volume 66, Issue 2(398), Pages 163–216 (Mi umn9420)

Arithmetic hypergeometric series

W. Zudilin

School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, Australia

Abstract: The main goal of this survey is to give common characteristics of auxiliary hypergeometric functions (and their generalisations), functions which occur in number-theoretic problems. Originally designed as a tool for solving these problems, the hypergeometric series have become a connecting link between different areas of number theory and mathematics in general.
Bibliography: 183 titles.

Keywords: hypergeometric series, zeta value, Ramanujan's mathematics, Diophantine approximation, irrationality measure, modular form, Calabi–Yau differential equation, Mahler measure, Wilf–Zeilberger theory, algorithm of creative telescoping.

DOI: https://doi.org/10.4213/rm9420

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English version:
Russian Mathematical Surveys, 2011, 66:2, 369–420

Bibliographic databases:

UDC: 511+517
MSC: Primary 33C20; Secondary 05A19, 11B65, 11F11, 11J82, 11M06, 11Y60, 14H52

Citation: W. Zudilin, “Arithmetic hypergeometric series”, Uspekhi Mat. Nauk, 66:2(398) (2011), 163–216; Russian Math. Surveys, 66:2 (2011), 369–420

Citation in format AMSBIB
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• https://doi.org/10.4213/rm9420
• http://mi.mathnet.ru/eng/umn/v66/i2/p163

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. M. Zakrzewski, H. Żołądek, “Linear differential equations and multiple zeta-values. III. Zeta(3)”, J. Math. Phys., 53:1 (2012), 013507, 40 pp.
2. J. Guillera, “More hypergeometric identities related to Ramanujan-type series”, Ramanujan J., 32:1 (2013), 5–22
3. P. B. Slater, “A concise formula for generalized two-qubit Hilbert–Schmidt separability probabilities”, J. Phys. A, 46:44 (2013), 445302, 13 pp.
4. E. A. Rakhmanov, S. P. Suetin, “The distribution of the zeros of the Hermite-Padé polynomials for a pair of functions forming a Nikishin system”, Sb. Math., 204:9 (2013), 1347–1390
5. M. A. Papanikolas, M. D. Rogers, D. Samart, “The Mahler measure of a Calabi-Yau threefold and special $L$-values”, Math. Z., 276:3-4 (2014), 1151–1163
6. E. Shinder, M. Vlasenko, “Mahler measures and double L-values of modular forms”, J. Number Theory, 142 (2014), 149–182
7. W. Zudilin, “Two hypergeometric tales and a new irrationality measure of $\zeta(2)$”, Ann. Math. Qué., 38:1 (2014), 101–117
8. S. Dauguet, W. Zudilin, “On simultaneous diophantine approximations to $\zeta(2)$ and $\zeta(3)$”, J. Number Theory, 145 (2014), 362–387
9. E. A. Karatsuba, “On One method for constructing a family of approximations of zeta constants by rational fractions”, Problems Inform. Transmission, 51:4 (2015), 378–390
10. Bartlett N., Warnaar S.O., “Hall–Littlewood polynomials and characters of affine Lie algebras”, Adv. Math., 285 (2015), 1066–1105
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