Several continued fraction expansions of generalized Cauchy numbers
type: Item
instance of: scholarly article
title: Several continued fraction expansions of generalized Cauchy numbers
zbMATH Open document ID: 1486.11009
author: Yanyan Li
published in: Bulletin of the Malaysian Mathematical Sciences Society. Second Series
publication date: 2021-08-12T00:00:00Z
review text: The author studies generalizations of the ordinary Cauchy numbers of the first kind \(c_n\) \[\frac{x}{\log{(1+x)}}=\sum_{n=0}^{\infty}\,c_n\,\frac{x^n}{n!}\tag{*}\] and several known continued fractions of the exponential generating function given above. The first generalization concerns so-called \textit{hypergeometric Cauchy numbers} \(c_{N,n}\) with \(N\geq 1\), introduced by the author [Int. J. Number Theory 9, No. 3, 545--560 (2013; Zbl 1301.11030)] is given by \[\frac{(-1)^Nx^N/N}{\log{(1+x)}-\sum_{n=1}^{N-1}\,(-1)^{n-1}x^n/n}=\sum_{n=0}^{\infty}\,c_{N,n}\,\frac{x^n}{n!},\tag{1}\] which reduces to \((*)\) for \(N=1\). The second generalization introduces the \textit{shifted Cauchy numbers (of the first kind)} \(c_n^{(m)}\) for non-negative \(n,m\), introduced by the author [``Shifted Cauchy numbers'', Filomat 32, No. 18, 6167--6176 (2018; \url{doi:10.2298/FIL1818167K})] (Remark: the author refers to reference [\textit{T. Komatsu} and \textit{C. Pita-Ruiz}, Quaest. Math. 43, No. 2, 213--226 (2020; Zbl 1444.11041)] here, but that concerns the leaping variant) \[ \frac{-1)^{m-1}x^m}{(-1)^{m-1}x^m +\log{(1+x)}-F_m(x)}=\sum_{n=0}^{\infty}\,c_n^{(m)}\,\frac{x^n}{n!},\tag{2}\] where \[F_m(x)=\sum_{n=1}^m\,(-1)^{n-1}\,\frac{x^n}{n}\ (m\geq 1)\tag{2a}\] Remark: Indeed, putting \(N=m-1\) in formula \((1)\) we find \((2), (2a)\), one can see the shift. The \textit{shifted Cauchy numbers (of the second kind)} \(\hat{c}_n^{(m)}\ n\geq 0,m\geq 1)\) are defined as in \((2)\), where in the denominator of the left hand side the extra terms \(x(\log{(1+x)}-F_{m-1}(x))\) are added. Finally the \textit{leaping Cauchy numbers} \(U_n^{(m)}\) (here the author refers to [8], which should be [\textit{T. Komatsu} and \textit{C. Pita-Ruiz}, Quaest. Math. 43, No. 2, 213--226 (2020; Zbl 1444.11041)]) are introduced by \[\sum_{n=0}^{\infty}\,U_n^{(m)}\,\frac{x^n}{n!}=\left(1-\frac{(-1)^{m/2}(\log{(1+x^2)}-F_{m/2}(x^2))}{2x^m}\right)^{-1}\] if \(m\) is even, where \(F_m(x)\) is defined in …
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 11A55
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 7381208
zbMATH Keywords: continued fractions
zbMATH Keywords: convergents
zbMATH Keywords: Cauchy numbers
zbMATH Keywords: hypergeometric Cauchy numbers
zbMATH Keywords: shifted Cauchy numbers
zbMATH Keywords: leaping Cauchy numbers
describes a project that uses: OEIS
MaRDI profile type: Publication
full work available at URL: s40840-021-01074-2
OpenAlex ID: W3125999533
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DOI: 10.1007/S40840-021-01074-2
description: scientific article; zbMATH DE number 7381208