Dougall 5F4求和公式的一些应用

阮玉盛

doi:10.3969/j.issn.1000-5641.2017.04.005

Dougall ₅F₄求和公式的一些应用

doi: 10.3969/j.issn.1000-5641.2017.04.005

阮玉盛

华东师范大学数学系, 上海 200241

基金项目:

国家自然科学基金 11571114

详细信息

作者简介:
阮玉盛, 男, 博士研究生, 研究方向为特殊函数与数论.E-mail:thinhnn02@yahoo.com

中图分类号: O156
计量
- 文章访问数: 209
- HTML全文浏览量: 67
- PDF下载量: 442
- 被引次数: 0
出版历程
- 收稿日期: 2016-10-19
- 刊出日期: 2017-07-25

Some applications of Dougall's ₅F₄ summation

NGUYEN Ngoc Thinh

Department of Mathematics, East China Normal University, Shanghai 200241, China

摘要

摘要: Dougall ₅F₄求和公式是特殊函数论中一个重要的级数求和公式，其在不同领域中的应用已被人们广泛讨论.本文以该公式为基础导出了一些新的求和公式，并利用这些公式给出了一系列新的关于1/π和1/π²的Ramanujan型级数公式.
- 伽马函数 /
- 超几何函数 /
- Dougall ₅F₄求和 /
- Ramanujan型级数
Abstract: Dougall's ₅F₄ summation formula plays an important role in the theory of special functions, and its various applications have been widely discussed. Using Dougall's ₅F₄ summation formula, we derive some new summation formulas, from which new Ramanujan type series for 1/π and 1/π² are obtained.
- gamma function /
- hypergeometric functions /
- Dougall's ₅F₄ summation /
- Ramanujan type series

HTML全文

参考文献(21)

[1]	RAMANUJAN S. Modular equations and approximations to [J]. Quart J Math Oxford Ser, 1914, 45(2): 350-372.
[2]	BORWEIN J M, BORWEIN P B. Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity [M]. New York: Wiley, 1987.
[3]	CHUDNOVSKY D V, CHUDNOVSKY G V, Approximations and complex multiplication according to Ramanu-jan [C]//Proceedings of the Centenary Conference, Urbana-Champaign, 1987. Boston: Academic Press, 1988, 375-472.
[4]	BARUAH N D, BERNDT B C. Ramanujan's series for 1/π arising from his cubic and quartic theories of elliptic functions [J]. J Math Anal Appl 2008, 341: 357-371. doi: 10.1016/j.jmaa.2007.10.011
[5]	BARUAH N D, BERNDT B C. Eisenstein Series and Ramanujan-type series for 1/π [J]. Ramanujan J, 2010, 23: 17-44. doi: 10.1007/s11139-008-9155-8
[6]	BARUAH N D, BERNDT B C, CHAN H H. Ramanujan's series for 1/π: A survey [J]. Amer Math Monthly, 2009, 116: 567-587. doi: 10.4169/193009709X458555
[7]	BARUAH N D, NAYAK N. New hypergeometric-like series for 1/π², arising from Ramanujan's theory of elliptic functions to alternative base 3 [J]. Trans Amer Math Soc, 2011, 363: 887-900. doi: 10.1090/S0002-9947-2010-05180-3
[8]	CHAN H H, CHAN S H, LIU Z G. Domb's numbers and Ramanujan-Sato type series for 1/π [J]. Adv in Math, 2004, 186: 396-410. doi: 10.1016/j.aim.2003.07.012
[9]	CHAN H H, COOPER S, LIAW W C. The Rogers-Ramanujan continued fraction and a quintic iteration for 1/π[J]. Proc Amer Math Soc, 2007, 135(11): 3417-3425. doi: 10.1090/S0002-9939-07-09031-4
[10]	CHAN H H, LIAW W C, TAN V. Ramanujan's class invariant n and a new class of series for 1/π [J]. J London Math Soc, 2001, 64(2): 93-106.
[11]	CHAN H H, LOO K L. Ramanujan's cubic continued revisited [J]. Acta Arith, 2007, 126: 305-313. doi: 10.4064/aa126-4-2
[12]	CHAN H H, VERRILL H. The Apéry numbers, the Almkvist-Zudilin numbers and new series for 1/π [J]. Math Res Lett, 2009, 16: 405-420. doi: 10.4310/MRL.2009.v16.n3.a3
[13]	CHAN H H, RUDILIN W. New representations for Apéry-like sequences 1/π [J]. Mathematika, 2010, 56: 107-117. doi: 10.1112/S0025579309000436
[14]	CHU W. Dougall's bilateral 2H₂ series and Ramanujan-like π formulas [J]. Math Comp, 2011, 80: 2223-2251. doi: 10.1090/S0025-5718-2011-02474-9
[15]	COOPER S. Series and iterations for 1/π [J]. Acta Arith, 2010, 141: 33-58. doi: 10.4064/aa141-1-2
[16]	GUILLERA J. Hypergeometric identities for 10 extended Ramanujan-type series [J]. Ramanujan J, 2008, 15:219-234. doi: 10.1007/s11139-007-9074-0
[17]	LEVRIE P. Using Fourier-Legendre expansions to derive series for 1/π and 1/π² [J]. Ramanujan J, 2010, 22:221-230. doi: 10.1007/s11139-010-9222-9
[18]	ROGERS M. New ₅F₄ hypergeometric transformations, three-variable Mahler measures and formulas for 1/π[J]. Ramanujan J, 2009, 18: 327-340. doi: 10.1007/s11139-007-9040-x
[19]	ZUDILIN W. More Ramanujan-type formulae for 1/π² [J]. Russian Math Surveys, 2007, 62 (3): 634-636. doi: 10.1070/RM2007v062n03ABEH004420
[20]	LIU Z G. A summation formula and Ramanujan type series [J]. J Math Anal App, 2012, 389: 1059-1065. doi: 10.1016/j.jmaa.2011.12.048
[21]	ANDREWS G E, ASKEY R, ROY R. Special Functions [M]. Cambridge: Cambridge University Press, 1999.