A new asymptotic expansion of a ratio of two gamma functions and complete monotonicity for its remainder
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- by Zhen-Hang Yang, Jing-Feng Tian and Ming-Hu Ha PDF
- Proc. Amer. Math. Soc. 148 (2020), 2163-2178 Request permission
Abstract:
In this paper, we establish a new asymptotic expansion of a ratio of two gamma functions, that is, as $x\rightarrow \infty$, \begin{equation*} \left [ \frac {\Gamma \left ( x+u\right ) }{\Gamma \left ( x+v\right ) }\right ] ^{1/\left ( u-v\right ) }\thicksim \left ( x+\sigma \right ) \exp \left [ \sum _{k=1}^{m}\frac {B_{2n+1}\left ( \rho \right ) }{wn\left ( 2n+1\right ) }\left ( x\!+\!\sigma \right ) ^{-2k}\!+\!R_{m}\left ( x;u,v\right ) \right ] , \end{equation*} 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.References
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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
- MR Author ID: 252484
- 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
- MR Author ID: 883754
- 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
- MR Author ID: 329090
- Email: mhhhbu@163.com
- 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
- Communicated by: Mourad Ismail
- © Copyright 2020 American Mathematical Society
- 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
- MathSciNet review: 4078101
Dedicated: Dedicated to the $60$th anniversary of Zhejiang Electric Power Company Research Institute