TY - JOUR
T1 - Asymptotics of Landau Constants with Optimal Error Bounds
AU - Li, Yutian
AU - Liu, Saiyu
AU - Xu, Shuaixia
AU - Zhao, Yuqiu
N1 - Funding information:
The authors are grateful to Prof. R. Wong for bringing the problem to their attention. The authors thank the anonymous referees for their careful reading of the manuscript and for their valuable suggestions and comments. One referee suggested using a quadratic transformation of the hypergeometric functions which makes the proof of Lemma 2 more natural and simplified. The other referee provided many constructive suggestions and corrections which have much improved the readability of the manuscript. The work of Yutian Li was supported in part by the HKBU Strategic Development Fund, a start-up Grant from Hong Kong Baptist University, and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKBU 201513). The work of Saiyu Liu was supported in part by Hunan Natural Science Foundation under Grant No. 14JJ6030 and by the National Natural Science Foundation of China under Grant No. 11326082. The work of Shuaixia Xu was supported in part by the National Natural Science Foundation of China under Grant No. 11201493, GuangDong Natural Science Foundation under Grant No. S2012040007824, and the Fundamental Research Funds for the Central Universities under Grant No. 13lgpy41. Yuqiu Zhao was supported in part by the National Natural Science Foundation of China under Grant Nos. 10471154 and 10871212.
Publisher copyright:
Copyright © 2014, Springer Science Business Media New York
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
UR - http://www.scopus.com/inward/record.url?scp=84908394783&partnerID=8YFLogxK
U2 - 10.1007/s00365-014-9259-x
DO - 10.1007/s00365-014-9259-x
M3 - Article
AN - SCOPUS:84908394783
VL - 40
SP - 281
EP - 305
JO - Constructive Approximation
JF - Constructive Approximation
SN - 0176-4276
IS - 2
ER -