Some recurrence relations of poly-Cauchy numbers

Abstract

Poly-Cauchy numbers c (k) 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 (−k) 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 (−k) 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.