On Cauchy Numbers and Their Generalizations

Abstract

This paper is concerned with both kinds of the Cauchy numbers and their generalizations. Takinginto account Mellin derivative, we relate 𝑝-Cauchy numbers of the second kind with shiftedCauchy numbers of the first kind, which yields new explicit formulas for the Cauchy numbers ofthe both kind. We introduce a generalization of the Cauchy numbers and investigate severalproperties, including recurrence relations, convolution identities and generating functions. Inparticular, these results give rise to new identities for Cauchy numbers.