华东师范大学学报（自然科学版） >

2017 >Issue 4: 52 - 63,70

DOI: https://doi.org/10.3969/j.issn.1000-5641.2017.04.005

数学

Dougall 5F4求和公式的一些应用

  • 阮玉盛
展开
  • 华东师范大学 数学系, 上海 200241
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.

收稿日期: 2016-10-19

  网络出版日期: 2017-07-20

基金资助

国家自然科学基金（11571114）

收起

Some applications of Dougall's 5F4 summation

  • NGUYEN Ngoc Thinh
Expand
  • Department of Mathematics, East China Normal University, Shanghai 200241, China

Received date: 2016-10-19

  Online published: 2017-07-20

Fold

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.

关键词： 伽马函数; 超几何函数; Dougall 5F4求和; Ramanujan 型级数

本文引用格式

阮玉盛 . Dougall 5F4求和公式的一些应用[J]. 华东师范大学学报（自然科学版）, 2017 , (4) : 52 -63,70 . DOI: 10.3969/j.issn.1000-5641.2017.04.005

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.

Key words： gamma function; hypergeometric functions; Dougall's 5F4 summation; Ramanujan type series

参考文献

[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.
[5] BARUAH N D, BERNDT B C. Eisenstein Series and Ramanujan-type series for 1/π [J]. Ramanujan J, 2010, 23: 17-44.
[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.
[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.
[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.
[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.
[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.
[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.
[13] CHAN H H, RUDILIN W. New representations for Apéry-like sequences 1/π [J]. Mathematika, 2010, 56: 107-117.
[14] CHU W. Dougall's bilateral 2H2 series and Ramanujan-like π formulas [J]. Math Comp, 2011, 80: 2223-2251.
[15] COOPER S. Series and iterations for 1/π [J]. Acta Arith, 2010, 141: 33-58.
[16] GUILLERA J. Hypergeometric identities for 10 extended Ramanujan-type series [J]. Ramanujan J, 2008, 15:
219-234.
[17] LEVRIE P. Using Fourier-Legendre expansions to derive series for 1/π and 1/π2 [J]. Ramanujan J, 2010, 22:
221-230.
[18] ROGERS M. New 5F4 hypergeometric transformations, three-variable Mahler measures and formulas for 1/π
[J]. Ramanujan J, 2009, 18: 327-340.
[19] ZUDILIN W. More Ramanujan-type formulae for 1/π2 [J]. Russian Math Surveys, 2007, 62 (3): 634-636.
[20] LIU Z G. A summation formula and Ramanujan type series [J]. J Math Anal App, 2012, 389: 1059-1065.
[21] ANDREWS G E, ASKEY R, ROY R. Special Functions [M]. Cambridge: Cambridge University Press, 1999.
Options
文章导航

/

〈 〉