The Cauchy numbers

“…are sometimes called the Bernoulli numbers of the second kind (see, e.g., [1,17]). Such numbers have been studied by several authors (see [4,14,15,16,18]) because they are related to various special combinatorial numbers, including Stirling numbers of both kinds, Bernoulli numbers, and harmonic numbers. The poly-Cauchy numbers c …”

Section: Introductionmentioning

confidence: 99%

Shifted poly-Cauchy numbers

Komatsu

Szalay

2014

Lith Math J

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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

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“…240-251], [136], [90, p. 132 [30,82,191,6,192,28], [124,Eq. (3)], [123,129], [9, pp. 128-129], [11,Chapt.…”

Section: Iii1 Introductionmentioning

confidence: 99%

Corrigendum to “Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only” [J. Number Theory 158 (2016) 365–396]

Blagouchine

2017

Journal of Number Theory

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In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) γ m are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in π −2 with rational coefficients and converges slightly better than Euler's series ∑ n −2 . The second expansion is a semi-convergent series with rational coefficients only. This expansion is particularly simple and involves Bernoulli numbers with a non-linear combination of generalized harmonic numbers. It also permits to derive an interesting estimation for generalized Euler's constants, which is more accurate than several well-known estimations. Finally, in Appendix A, the reader will also find two simple integral definitions for the Stirling numbers of the first kind, as well an upper bound for them.

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The Cauchy numbers

Cited by 99 publications

References 6 publications

Convolution Identities for Cauchy Numbers of the Second Kind

Convolution Identities for Cauchy Numbers of the Second Kind

Shifted poly-Cauchy numbers

Corrigendum to “Expansions of generalized Euler's constants into the series of polynomials in π−2 and into the formal enveloping series with rational coefficients only” [J. Number Theory 158 (2016) 365–396]

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