Some recurrence relations of poly-Cauchy numbers

Poly-Cauchy numbers c n (n > 0, k > 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 }n>0 seem quite irregular for a fixed integer k > 2. In this paper we establish a certain kind of recurrence relations among the sequence {c n }n>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.