# 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
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### Authors

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

### Abstract

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.

### Share and Cite

##### ISRP Style

Takao Komatsu, Some recurrence relations of poly-Cauchy numbers, Journal of Nonlinear Sciences and Applications, 12 (2019), no. 12, 829--845

##### AMA Style

Komatsu Takao, Some recurrence relations of poly-Cauchy numbers. J. Nonlinear Sci. Appl. (2019); 12(12):829--845

##### Chicago/Turabian Style

Komatsu, Takao. "Some recurrence relations of poly-Cauchy numbers." Journal of Nonlinear Sciences and Applications, 12, no. 12 (2019): 829--845

### Keywords

• Poly-Cauchy numbers
• poly-Euler numbers
• recurrence
• leaping relations
• Vandermonde's determinant

•  11B75
•  11B37
•  11B68
•  11B73
•  05A19
•  11C20
•  15A15

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