Asymptotic expansions of Gurland's ratio and sharp bounds for their remainders |
| |
Authors: | Jing-Feng Tian Zhenhang Yang |
| |
Affiliation: | 1. Department of Mathematics and Physics, North China Electric Power University, Yonghua Street 619, 071003, Baoding, PR China;2. Engineering Research Center of Intelligent Computing for Complex Energy Systems of Ministry of Education, North China Electric Power University, Yonghua Street 619, 071003, Baoding, PR China |
| |
Abstract: | In this paper, we establish a new asymptotic expansion of Gurland's ratio of gamma functions, that is, as ,where with and , , are the Bernoulli polynomials. Using a double inequality for hyperbolic functions, we prove that the function is completely monotonic on if , which yields a sharp upper bound for . This shows that the approximation for Gurland's ratio by the truncation of the above asymptotic expansion has a very high accuracy. We also present sharp lower and upper bounds for Gurland's ratio in terms of the partial sum of hypergeometric series. Moreover, some known results are contained in our results when . |
| |
Keywords: | Gurland's ratio of gamma function Asymptotic expansion Complete monotonicity Hypergeometric series |
本文献已被 ScienceDirect 等数据库收录! |
|