arXiv subjects
Publications and source records attributed to C. -P. Chen.
In this paper, we present series representations of the remainders in the expansions for certain trigonometric and hyperbolic functions. By using the obtained results, we establish some inequalities for trigonometric and hyperbolic functions.
In this paper, we present series representations of the remainders in the expansions for $2/(e^t+1)$, $\mbox{sech} t$ and $\coth t$. For example, we prove that for $t > 0$ and $N\in\mathbb{N}:=\{1, 2, \ldots\}$, \[\mbox{sech}\, t=\sum_{j=0}^{N-1}\frac{E_{2j}}{(2j)!}t^{2j}+R_N(t) \] with \[ R_N(t)=\frac{(-1)^{N}2t^{2N}}{π^{2N-1}}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(k+\frac{1}{2})^{2N-1}\Big(t^2+π^2(k+\frac{1}{2})^2\Big)}, \] and \[\mbox{sech}\, t=\sum_{j=0}^{N-1}\frac{E_{2j}}{(2j)!}t^{2j}+Θ(t, N)\frac{E_{2N}}{(2N)!}t^{2N} \] with a suitable $0 < Θ(t, N) < 1$. Here $E_n$ are the Euler numbers. By using the obtained results, we deduce some inequalities and completely monotonic functions associated with the ratio of gamma functions. Furthermore, we give a (presumably new) quadratic recurrence relation for the Bernoulli numbers.
For all integers $n\geq1$, let \begin{align*} W_n(p,q)=\prod_{j=1}^{n}\left\{e^{-p/j}\left(1+\frac{p}{j}+\frac{q}{j^2}\right)\right\} \end{align*} and \begin{align*} R_n(p, q)=\prod_{j=1}^{n}\left\{e^{-p/(2j-1)}\left(1+\frac{p}{2j-1}+\frac{q}{(2j-1)^2}\right)\right\}, \end{align*} where $p$, $q$ are complex parameters. The infinite product $W_{\infty}(p,q)$ includes the Wallis and Wilf formulas, and also the infinite product definition of Weierstrass for the gamma function, as special cases. In this paper, we present asymptotic expansions of $W_n(p,q)$ and $R_n(p, q)$ as $n\to\infty$. In addition, we also establish asymptotic expansions for the Wallis sequence.
We report the latest results of 225 GHz atmospheric opacity measurements from two arctic sites; one on high coastal terrain near the Eureka weather station, on Ellesmere Island, Canada, and the other at the Summit Station near the peak of the Greenland icecap. This is a campaign to search for a site to deploy a new telescope for submillimeter Very Long Baseline Interferometry and THz astronomy in the northern hemisphere. Since 2011, we have obtained 3 months of winter data near Eureka, and about one year of data at the Summit Station. The results indicate that these sites offer a highly transparent atmosphere for observations in submillimeter wavelengths. The Summit Station is particularly excellent, and its zenith opacity at 225 GHz is statistically similar to the Atacama Large Milllimeter/submillimeter Array in Chile. In winter, the opacity at the Summit Station is even comparable to that observed at the South Pole.