Several continued fraction expansions of generalized Cauchy numbers
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 \((2a)\), and \[\sum_{n=0}^{\infty}\,U_n^{(m)}\,\frac{x^n}{n!}=\left(1+\frac{(-1)^{(m-1)/2}(\arctan{x}-F_{odd,(m+1)/2}(x))}{x^m}\right)^{-1}\] if \(m\) is odd, where \[F_{odd,m}(x)=\sum_{n=1}^m\,(-1)^{n-1}\,\frac{x^{2n-1}}{2n-1}.\] The layout of the paper is now as follows: \S1 Introduction (\(2\frac{1}{2}\) pages). \S2 Continued fractions (\(1\) page). \S3 Hypergeometric Cauchy numbers (\(2\frac{1}{2}\) pages). It gives a new continued fraction (Theorem 1) for the generating function. \S4 Shifted Cauchy Numbers (\(6\) pages). It states new results in Theorems 2 and 3. \S5 Shifted Cauchy numbers of the second kind (\(4\frac{1}{2}\) pages). There are new results in Theorems 4 and 5. \S6 Leaping Cauchy numbers (\(4\) pages). It states Theorems 6 and 7. \S7 Final comments. References (\(16\) items).
scientific article; zbMATH DE number 7381208