Department of Mathematics
Asymptotics of landau constants with optimal error bounds
© 2014, Springer Science+Business Media New York.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).
Asymptotic expansion, Landau constants, Second-order linear difference equation, Sharper bound
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Li, Y., Liu, S., Xu, S., & Zhao, Y. (2014). Asymptotics of landau constants with optimal error bounds. Constructive Approximation, 40 (2), 281-305. https://doi.org/10.1007/s00365-014-9259-x