Asymptotics of Landau Constants with Optimal Error Bounds

Yutian LI; Saiyu Liu; Shuaixia Xu; Yuqiu Zhao

doi:10.1007/s00365-014-9259-x

Asymptotics of Landau Constants with Optimal Error Bounds

Yutian LI, Saiyu Liu, Shuaixia Xu, Yuqiu Zhao^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

We study the asymptotic expansion for the Landau constants G_n, (Formula Presented) where N=n+3/4, γ =0.5772 … is Euler’s constant, and (-1)^s+1β_2sare positive rational numbers, given explicitly in an iterative manner. We show that the error due to truncation is bounded in absolute value by, and of the same sign as, the first neglected term for all nonnegative n. Consequently, we obtain optimal sharp bounds up to arbitrary orders of the form (Formula Presented) for all n = 0,1,2,…, m=1,2,…, and k = 1,2,…. The results are proved by approximating the coefficients β_2swith the Gauss hypergeometric functions involved and by using the second-order difference equation satisfied by G_n, as well as an integral representation of the constants (Formula Presented).

Original language	English
Pages (from-to)	281-305
Number of pages	25
Journal	Constructive Approximation
Volume	40
Issue number	2
DOIs	https://doi.org/10.1007/s00365-014-9259-x
Publication status	Published - Oct 2014

Scopus Subject Areas

Analysis
Mathematics(all)
Computational Mathematics

User-Defined Keywords

Asymptotic expansion
Landau constants
Second-order linear difference equation
Sharper bound

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10.1007/s00365-014-9259-x

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title = "Asymptotics of Landau Constants with Optimal Error Bounds",

abstract = "We study the asymptotic expansion for the Landau constants Gn, (Formula Presented) where N=n+3/4, γ =0.5772 … is Euler{\textquoteright}s constant, and (-1)s+1β2sare positive rational numbers, given explicitly in an iterative manner. We show that the error due to truncation is bounded in absolute value by, and of the same sign as, the first neglected term for all nonnegative n. Consequently, we obtain optimal sharp bounds up to arbitrary orders of the form (Formula Presented) for all n = 0,1,2,…, m=1,2,…, and k = 1,2,…. The results are proved by approximating the coefficients β2swith the Gauss hypergeometric functions involved and by using the second-order difference equation satisfied by Gn, as well as an integral representation of the constants (Formula Presented).",

keywords = "Asymptotic expansion, Landau constants, Second-order linear difference equation, Sharper bound",

author = "Yutian LI and Saiyu Liu and Shuaixia Xu and Yuqiu Zhao",

year = "2014",

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doi = "10.1007/s00365-014-9259-x",

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journal = "Constructive Approximation",

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T1 - Asymptotics of Landau Constants with Optimal Error Bounds

AU - LI, Yutian

AU - Liu, Saiyu

AU - Xu, Shuaixia

AU - Zhao, Yuqiu

PY - 2014/10

Y1 - 2014/10

N2 - We study the asymptotic expansion for the Landau constants Gn, (Formula Presented) where N=n+3/4, γ =0.5772 … is Euler’s constant, and (-1)s+1β2sare positive rational numbers, given explicitly in an iterative manner. We show that the error due to truncation is bounded in absolute value by, and of the same sign as, the first neglected term for all nonnegative n. Consequently, we obtain optimal sharp bounds up to arbitrary orders of the form (Formula Presented) for all n = 0,1,2,…, m=1,2,…, and k = 1,2,…. The results are proved by approximating the coefficients β2swith the Gauss hypergeometric functions involved and by using the second-order difference equation satisfied by Gn, as well as an integral representation of the constants (Formula Presented).

AB - We study the asymptotic expansion for the Landau constants Gn, (Formula Presented) where N=n+3/4, γ =0.5772 … is Euler’s constant, and (-1)s+1β2sare positive rational numbers, given explicitly in an iterative manner. We show that the error due to truncation is bounded in absolute value by, and of the same sign as, the first neglected term for all nonnegative n. Consequently, we obtain optimal sharp bounds up to arbitrary orders of the form (Formula Presented) for all n = 0,1,2,…, m=1,2,…, and k = 1,2,…. The results are proved by approximating the coefficients β2swith the Gauss hypergeometric functions involved and by using the second-order difference equation satisfied by Gn, as well as an integral representation of the constants (Formula Presented).

KW - Asymptotic expansion

KW - Landau constants

KW - Second-order linear difference equation

KW - Sharper bound

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DO - 10.1007/s00365-014-9259-x

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JO - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

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ER -

Asymptotics of Landau Constants with Optimal Error Bounds

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