Completely monotonic integer degrees for a class of special functions

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 $\left(-1\right)^{m}R_{n}^{(m)}(x)$...

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Bibliographic Details
Published inAIMS mathematics Vol. 5; no. 4; pp. 3456 - 3471
Main Author Zhu, Ling
Format Journal Article
LanguageEnglish
Published AIMS Press 01.01.2020
Subjects
completely monotonic function
completely monotonic integer degree
gamma function
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Summary: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 $m=0,1,2,\cdots$.
ISSN:2473-6988
2473-6988
DOI:10.3934/math.2020224