Czechoslovak Mathematical Journal, Vol. 74, No. 1, pp. 337-351, 2024
Complete monotonicity of the remainder in an asymptotic series related to the psi function
Zhen-Hang Yang, Jing-Feng Tian
Received August 2, 2023. Published online February 12, 2024.
Abstract: Let $p,q\in\mathbb{R}$ with $p-q\geq0$, $\sigma= \frac12 (p+q-1)$ and $s=\frac12 (1-p+q)$, and let $\mathcal{D}_m (x;p,q) =\mathcal{D}_0 ( x;p,q ) +\sum_{k=1}^m\frac{B_{2k} ( s) }{2k (x+\sigma)^{2k}},$ where $\mathcal{D}_0 (x;p,q) =\frac{\psi(x+p) +\psi(x+q) }2-\ln(x+\sigma).$ We establish the asymptotic expansion $\mathcal{D}_0 (x;p,q) \sim-\sum_{n=1}^{\infty} \frac{B_{2n} (s)}{2n (x+\sigma) ^{2n}}$ as $x\rightarrow\infty,$ where $B_{2n} (s)$ stands for the Bernoulli polynomials. Further, we prove that the functions $(-1) ^m\mathcal{D}_m ( x;p,q )$ and $(-1) ^{m+1}\mathcal{D}_m ( x;p,q )$ are completely monotonic in $x$ on $( -\sigma,\infty)$ for every $m\in\mathbb{N}_0$ if and only if $p-q\in[ 0, \tfrac12 ]$ and $p-q=1$, respectively. This not only unifies the two known results but also yields some new results.
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Affiliations: Zhen-Hang Yang, State Grid Zhejiang Electric Power Company Research Institute, Hangzhou, Zhejiang 310014, P. R. China, e-mail: yzhkm@163.com; Jing-Feng Tian (corresponding author), Hebei Key Laboratory of Physics and Energy Technology, Department of Mathematics and Physics, North China Electric Power University, Baoding, Hebei 071003, P. R. China, e-mail: tianjf@ncepu.edu.cn
