On s-Stirling transform and poly-Cauchy numbers of the second kind with level 2

In this paper, we study the transform by Stirling numbers with higher level, and give several concrete results. When s=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=2$$\end{document}, we consider the transform of rational sequences. In particular, poly-Cauchy numbers of the second kind with level 2 are introduced in order to achieve some extended results. We also give several properties of poly-Cauchy numbers of the second kind with level 2, which are related to those of poly-Bernoulli numbers with level 2 and analogous to those of poly-Cauchy numbers of the first kind with level 2.