PSI, POLYGAMMA FUNCTIONS AND <i>Q</i>-COMPLETE MONOTONICITY ON TIME SCALES

Zhong-Xuan Mao; Jing-Feng Tian; Ya-Ru Zhu

doi:10.11948/20210340

2023 Volume 13 Issue 3

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Article navigation > Journal of Applied Analysis & Computation > 2023 > 13(3): 1137-1154

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Zhong-Xuan Mao, Jing-Feng Tian, Ya-Ru Zhu. PSI, POLYGAMMA FUNCTIONS AND Q-COMPLETE MONOTONICITY ON TIME SCALES[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1137-1154. doi: 10.11948/20210340

Citation:

Zhong-Xuan Mao, Jing-Feng Tian, Ya-Ru Zhu. PSI, POLYGAMMA FUNCTIONS AND Q-COMPLETE MONOTONICITY ON TIME SCALES[J]. Journal of Applied Analysis & Computation, 2023, 13(3): 1137-1154. doi: 10.11948/20210340

PSI, POLYGAMMA FUNCTIONS AND Q-COMPLETE MONOTONICITY ON TIME SCALES

1.
Department of Mathematics and Physics, North China Electric Power University, Yonghua Street 619, 071003 Baoding, China

Author Bio: Email: maozhongxuan@ncepu.edu.cn(Z. X. Mao); Email: zhuyaru1982@ncepu.edu.cn(Y. R. Zhu)
Corresponding author: Email: tianjf@ncepu.edu.cn(J. F. Tian)

Abstract

Abstract

In this paper, we generalize psi and polygamma functions based on the Laplace transform in the field of time scales, and explore some properties of them. Next, we present the concepts of $ q $-complete monotonicity, $ q $-logarithmically complete monotonicity and $ q $-absolute monotonicity with delta derivative on time scales. At last, we prove that the function

$ \begin{equation*} s\mapsto \alpha \psi_{\mathbb{R}_0, \mathbb{T}}(s)-\ln s+\frac{1}{2s}+\frac{1}{12s^2} \end{equation*} $

is $ 1 $-complete monotonicity on $ (0, \infty) $ if $ \mathbb{T}=\mathbb{N} $ and $ \alpha \in [\frac{3-2\sqrt{3}}{6}, \frac{3+2\sqrt{3}}{6}] $, and it is decreasing on $ (0, \infty) $ if $ \mathbb{T}=h\mathbb{N}\cup\{1\} (h\geq1) $ and $ \alpha=1 $, where $ \mathbb{R}_0=[0, \infty) $ and $ \psi_{\mathbb{R}_0, \mathbb{T}} $ is a psi function on time scales.
- Psi function /
- polygamma function /
- completely monotonic /
- gamma function /
- time scale
MSC: 33B15, 26A48, 26E70

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References

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