AIMS Mathematics, 2020, 5(4): 3456-3471. doi: 10.3934/math.2020224.

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Completely monotonic integer degrees for a class of special functions

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

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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Keywords completely monotonic function; completely monotonic integer degree; gamma function

Citation: Ling Zhu. Completely monotonic integer degrees for a class of special functions. AIMS Mathematics, 2020, 5(4): 3456-3471. doi: 10.3934/math.2020224

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