Completely monotonic integer degrees for a class of special functions

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 $m=0,1,2,\cdots$.

Keywords

• completely monotonic function
• completely monotonic integer degree
• gamma function