In the present article we discuss necessary and sufficient conditions for concave functions, i.e. meromorphic functions which map the unit disk conformally on a domain whose complement is convex. The conditions will be given with respect to an arbitrary point $p\in (-1,1)$. We will also look at representation formulas for the related functions as well as an application of the derived formula.

T. Agoh and K. Dilcher proved convolution identities of Bernoulli numbers in 2007. Their proof was complicated calculations in more than 10 pages, which were based on the relation between the Stirling numbers of second kind and the Bernoulli numbers. In this short paper, we give a simple proof of it. Essentially, the proof is based on just one formula on a new kind of generating function.

This article concerns with the first Painlevé hierarchy, i.e. the $2n$-th order analogues of the first Painlevé equation. Several higher order analogues of the first Painlevé equation are proposed by several authors however, we investigate one derived from the KdV hierarchy by studying singularity manifold. We give the auxiliary differential polynomials of the first Painlevé hierarchy.

For a one dimensional diffusion process $\mathbf{D}^{*}_{s,m}$ and the harmonic transformed process $\mathbf{D}^{*}_{s_{h},m_{h}}$, the asymptotic behavior of the Lévy measure density corresponding to the inverse local time at the regular end point is investigated. The asymptotic behavior of $n^{*}$, the Lévy measure density corresponding to $\mathbf{D}^{*}_{s,m}$, follows from asymptotic behavior of the speed measure $m$. However, that of $n^{h*}$, the Lévy measure density corresponding to $\mathbf{D}^{*}_{s_{h},m_{h}}$, is given by a simple form, $n^{*}$ multiplied by an exponential decay function, for any harmonic function $h$ based on the original diffusion operator.