Asymptotics of Landau Constants with Optimal Error Bounds

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

Abstract

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).

Original languageEnglish
Pages (from-to)281-305
Number of pages25
JournalConstructive Approximation
Volume40
Issue number2
DOIs
Publication statusPublished - 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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