Completely monotonic integer degrees for a class of special functions

Ling Zhu; Ling Zhu

doi:10.3934/math.2020224

AIMS Mathematics

2020, Volume 5, Issue 4: 3456-3471. doi: 10.3934/math.2020224

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Research article

Completely monotonic integer degrees for a class of special functions

Ling Zhu ^,

Department of Mathematics, Zhejiang Gongshang University, Hangzhou 310018, P. R. China

Received: 14 January 2020 Accepted: 26 March 2020 Published: 07 April 2020
MSC : 26A48, 33B15, 44A10

Let $f_{n}(x)$ $\left(n = 0, 1, \cdots\right)$ be the remainders for the asymptotic formula of $\ln\Gamma (x)$ and $R_{n}(x) = \left(-1\right)^{n}f_{n}(x)$. This paper introduced the concept of completely monotonic integer degree and discussed the ones for the functions $\left(-1\right)^{m}R_{n}^{(m)}(x)$, then demonstrated the correctness of the existing conjectures by using a elementary simple method. Finally, we propose some operational conjectures which involve the completely monotonic integer degrees for the functions $\left(-1\right) ^{m}R_{n}^{(m)}(x)$ for $m = 0, 1, 2, \cdots$.
- completely monotonic function,
- completely monotonic integer degree,
- gamma function
Citation: Ling Zhu. Completely monotonic integer degrees for a class of special functions[J]. AIMS Mathematics, 2020, 5(4): 3456-3471. doi: 10.3934/math.2020224

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Abstract

Let $f_{n}(x)$ $\left(n = 0, 1, \cdots\right)$ be the remainders for the asymptotic formula of $\ln\Gamma (x)$ and $R_{n}(x) = \left(-1\right)^{n}f_{n}(x)$. This paper introduced the concept of completely monotonic integer degree and discussed the ones for the functions $\left(-1\right)^{m}R_{n}^{(m)}(x)$, then demonstrated the correctness of the existing conjectures by using a elementary simple method. Finally, we propose some operational conjectures which involve the completely monotonic integer degrees for the functions $\left(-1\right) ^{m}R_{n}^{(m)}(x)$ for $m = 0, 1, 2, \cdots$.

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