Some recurrence relations of poly-Cauchy numbers

Volume 12, Issue 12, pp 829--845
Publication Date: August 07, 2019 Submission Date: June 01, 2019 Revision Date: July 01, 2019 Accteptance Date: July 23, 2019


Takao Komatsu - Department of Mathematical Sciences, School of Science, Zhejiang Sci-Tech University, Hangzhou 310018, China.


Poly-Cauchy numbers \(c_n^{(k)}\) (\(n\ge 0\), \(k\ge 1\)) have explicit expressions in terms of the Stirling numbers of the first kind. When the index is negative, there exists a different expression. However, the sequence \(\{c_n^{(-k)}\}_{n\ge 0}\) seem quite irregular for a fixed integer \(k\ge 2\). In this paper we establish a certain kind of recurrence relations among the sequence \(\{c_n^{(-k)}\}_{n\ge 0}\), analyzing the structure of poly-Cauchy numbers. We also study those of poly-Cauchy numbers of the second kind, poly-Euler numbers, and poly-Euler numbers of the second kind. Some different proofs are given. As applications, some leaping relations are shown.