Landau常数的逼近公式

数学物理学报 ›› 2014, Vol. 34 ›› Issue (5): 1245-1253.

Landau常数的逼近公式

陈超平, 成军祥

河南理工大学数学与信息科学学院 |河南焦作 454000

收稿日期:2013-10-15 修回日期:2014-05-07 出版日期:2014-10-25 发布日期:2014-10-25
基金资助:
国家自然科学基金(10471126, 10371109)、浙江省自然科学基金(101016)和浙江省哲学社会科学规划常规性课题(06CGYJ21YBQ)资助

Approximation Formulas for the Landau Constants

School of Mathematics and Informatics, Henan Polytechnic University, Henan Jiaozuo 454000

Received:2013-10-15 Revised:2014-05-07 Online:2014-10-25 Published:2014-10-25

摘要/Abstract

摘要：

对于所有的整数n ≥0, Landau常数定义为
G_n=∑_k=0ⁿ1/16^k(2k k}².
该文建立了Landau常数新的逼近公式. 指出了获得的结果与先前已有结果之间的相关联系.

关键词: Landau常数, Psi函数, 逼近

Abstract:

The Landau constants are defined for all integers n ≥0 by
G_n=∑_k=0ⁿ1/16^k(2k k)².
We establish new approximation formulas for the Landau constants. Relevant connections of the results presented here with those derived in earlier work are also pointed out.

Key words: Landau constants, Psi function, Approximation

中图分类号:

11Y60

陈超平, 成军祥. Landau常数的逼近公式[J]. 数学物理学报, 2014, 34(5): 1245-1253.

CHEN Chao-Ping, CHENG Jun-Xiang. Approximation Formulas for the Landau Constants[J]. Acta mathematica scientia,Series A, 2014, 34(5): 1245-1253.

参考文献

[1] Landau E. Absch\"{a}tzung der Koeffzientensumme einer Potenzreihe. Arch Math Phys, 1913, 21: 42--50, 250--255

[2] Watson G N. The constants of Landau and Lebesgue. Quart J Math Oxford Ser, 1930, 1: 310--318

[3] Brutman L. A sharp estimate of the Landau constants. J Approx Theory, 1982, 34(2): 217--220

[4] Falaleev L P. Inequalities for the Landau constants. Siberian Math J, 1991, 32(5): 896--897

[5] Chen C P. Approximation formulas for Landau's constants. J Math Anal Appl, 2012, 387(2): 916--919

[6] Chen C P. Sharp bounds for the Landau constants. Ramanujan J, 2013, 31(3): 301--313

[7] Cvijovi\'c D, Srivastava H M. Asymptotics of the Landau constants and their relationship with hypergeometric functions. Taiwanese J Math, 2009, 13(3): 855--870

[8] Granath H. On inequalities and asymptotic expansions for the Landau constants. J Math Anal Appl, 2012, 386(2): 738--743

[9] Li Y, Li S, Xu S, Zhao Y. Full asymptotic expansions of the Landau constants via a difference equation approach. Appl
Math Comput, 2012, 219(3): 988--995

[10] Nemes G. Proofs of two conjectures on the Landau constants. J Math Anal Appl, 2012, 388(2): 838--844

[11] Nemes G, Nemes A. A note on the Landau constants. Appl Math Comput, 2011, 217(21): 8543--8546

[12] Mortici C. Sharp bounds of the Landau constants. Math Comput, 2011, 80(274): 1011--1018

[13] Popa E C. Note of the constants of Landau. Gen Math, 2010, 18(1): 113--117

[14] Popa E C, Secelean N A. On some inequality for the Landau constants. Taiwan J Math, 2011, 15(4): 1457--1462

[15] Zhao D. Some sharp estimates of the constants of Landau and Lebesque. J Math Anal Appl, 2009, 349(1): 68--73

[16] Abramowitz M, Stegun I A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Applied Mathematics. Series 55. New York: Dover, 1972

[17] Cvijovi\'c D, Klinowski J. Inequalities for the Landau constants. Math Slovaca, 2000, 50(2): 159--164

[18] Alzer H. Inequalities for the constants of Landau and Lebesgue. J Comput Appl Math, 2002, 139(2): 215--230

[19] Mortici C. New approximations of the gamma function in terms of the digamma function. Appl Math Lett, 2010, 23(1): 97--100

[20] Mortici C. Product approximations via asymptotic integration. Amer Math Monthly, 2010, 117(5): 434--441

[21] Mortici C. On new sequences converging towards the Euler-Mascheroni constant. Comput Math Appl, 2010, 59(8): 2610--2614

[22] Mortici C. A quicker convergence toward the ε constant with the logarithm term involving the constant e. Carpathian J Math, 2010, 26(1): 86--91