Some recurrence relations of poly-Cauchy numbers

PolyCauchy 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 polyCauchy numbers. We also study those of polyCauchy numbers of the second kind, polyEuler numbers, and polyEuler numbers of the second kind. Some different proofs are given.As applications, some leaping relations are shown.