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A new asymptotic expansion of a ratio of two gamma functions and complete monotonicity for its remainder


Authors: Zhen-Hang Yang, Jing-Feng Tian and Ming-Hu Ha
Journal: Proc. Amer. Math. Soc. 148 (2020), 2163-2178
MSC (2010): Primary 41A60, 33B15; Secondary 26A48, 26D15
DOI: https://doi.org/10.1090/proc/14917
Published electronically: January 28, 2020
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Abstract: In this paper, we establish a new asymptotic expansion of a ratio of two gamma functions, that is, as $ x\rightarrow \infty $,

$\displaystyle \left [ \frac {\Gamma \left ( x+u\right ) }{\Gamma \left ( x+v\ri... ...}\left ( x\!+\!\sigma \right ) ^{-2k}\!+\!R_{m}\left ( x;u,v\right ) \right ] ,$    

where $ u,v\in \mathbb{R}$ with $ w=u-v\neq 0$ and $ \rho =\left ( 1-w\right ) /2$, $ \sigma =\left ( u+v-1\right ) /2$, $ B_{2n+1}\left ( \rho \right ) $ are the Bernoulli polynomials. We also prove that the function $ x\mapsto \left ( -1\right ) ^{m}R_{m}\left ( x;u,v\right ) $ for $ m\in \mathbb{N}$ is completely monotonic on $ \left ( -\sigma ,\infty \right ) $ if $ \left \vert u-v\right \vert <1$, which yields an explicit bound for $ \left \vert R_{m}\left ( x;u,v\right ) \right \vert $ and some new inequalities.

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Additional Information

Zhen-Hang Yang
Affiliation: Engineering Research Center of Intelligent Computing for Complex Energy Systems of Ministry of Education, North China Electric Power University, Yonghua Street 619, 071003, Baoding, People’s Republic China; and Zhejiang Electric Power Society, Hangzhou, Zhejiang, 310008, People’s Republic of China
Email: yzhkm@163.com

Jing-Feng Tian
Affiliation: Department of Mathematics and Physics, North China Electric Power University, Yonghua Street 619, 071003, Baoding, People’s Republic of China
Email: tianjf@ncepu.edu.cn

Ming-Hu Ha
Affiliation: School of Science, Hebei University of Engineering, Handan, Hebei Province, 056038, People’s Republic of China
Email: mhhhbu@163.com

DOI: https://doi.org/10.1090/proc/14917
Keywords: Ratio of gamma function, remainder of asymptotic expansion, completely monotonicity, inequality
Received by editor(s): June 28, 2019
Received by editor(s) in revised form: October 11, 2019
Published electronically: January 28, 2020
Additional Notes: The third author is the corresponding author
Dedicated: Dedicated to the $60$th anniversary of Zhejiang Electric Power Company Research Institute
Communicated by: Mourad Ismail
Article copyright: © Copyright 2020 American Mathematical Society