Abstract
In Komatsu's work (2013), the concept of poly-Cauchy numbers is introduced as an analogue of that of poly-Bernoulli numbers. Both numbers are extensions of classical Cauchy numbers and Bernoulli numbers, respectively. There are several generalizations of poly-Cauchy numbers, including poly-Cauchy numbers with a q parameter and shifted poly-Cauchy numbers. In this paper, we give a further generalization of poly-Cauchy numbers and investigate several arithmetical properties. We also give the corresponding generalized poly-Bernoulli numbers so that both numbers have some relations.
Highlights
IntroductionPoly-Cauchy numbers of the first kind cn(k) are defined by cn(k)
If q = l = 1, Theorem 3 is reduced to Theorem 3 in [8]
It is known that poly-Bernoulli numbers satisfy the duality theorem Bn(−k) = Bk(−n) for n, k ≥ 0 ([6], Theorem 2) because of the symmetric formula
Summary
Poly-Cauchy numbers of the first kind cn(k) are defined by cn(k). The concept of poly-Cauchy numbers is a generalization of that of the classical Cauchy numbers cn = cn(1) defined by cn = ∫ x (x − 1) ⋅ ⋅ ⋅ (x − n + 1) dx (2). The generating function of poly-Cauchy numbers ([1], Theorem 2) is given by. When k = 1, Bn = Bn(1) is the classical Bernoulli number with B1(1) = 1/2, defined by the generating function xex ex − 1. Remember that the Hurwitz zeta function ζ(s, q) = Riemann z∑et∞ na=0fu1n/c(qtio+nnζ)(ss)is=a∑g∞ ne=n1e1r/anliszasitniocne of the ζ(s) =. Cauchy numbers, including both kinds of generalizations, and show several combinatorial and characteristic properties. We give the corresponding poly-Bernoulli numbers so that both numbers have some relations
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