Shifted poly-Cauchy numbers

Abstract

Abstract. Recently, the first author introduced the concept of poly-Cauchy numbers as a generalization of the classical Cauchy numbers and an analogue of poly-Bernoulli numbers. This concept has been generalized in various ways, including poly-Cauchy numbers with a q parameter. In this paper, we give a different kind of generalization called shifted poly-Cauchy numbers and investigate several arithmetical properties. Such numbers can be expressed in terms of original poly-Cauchy numbers. This concept is a kind of analogous ideas to that of Hurwitz zeta-functions compared to Riemann zeta-functions. MSC: 05A15, 11B73, 11B75, 11B83

Generalizations of poly-Bernoulli and poly-Cauchy numbers

“…In [9], poly-Bernoulli numbers are generalized with a q parameter. In [25], shifted poly-Cauchy and poly-Bernoulli numbers are defined and in [23] these numbers are further generalized with a q parameter. In [12,16], poly-Bernoulli and poly-Cauchy numbers and polynomials are considered by means of multiparameters.…”

“…We remark that the Hurwitz type poly-Cauchy numbers are defined as shifted poly-Cauchy numbers in [25] and a generalization is given in [23]. Komatsu [22] defined the poly-Cauchy numbers of the second kind…”

A Generalization of Poly-Cauchy Numbers and Their Properties

“…If = ℓ = 1, then Theorem 8 is reduced to Theorem 3 in [7] and Theorem 9 is reduced to Theorem 4 in [7]. If = ℓ = 1, then Theorem 8 is reduced to Theorem 5 in [8] and Theorem 9 is reduced to Theorem 6 in [8].…”

“…If = ℓ = 1, then Proposition 15 is reduced to Proposition 1 in [7]. If = ℓ = 1, then Proposition 15 is reduced to Proposition 3 in [8].…”

On Cauchy Numbers and Their Generalizations

“…̂) is ( ) ( ), called shifted poly-CNFK, (resp. ̂( ) ( ) shifted poly-CNSK) and defined by [13] ( ) ( ) = ∫…”