The central Bell polynomials B (c) k (x) for k ≥ 0 can be generated [5] by exp 2x sinh t 2 = X∞ k=0 B (c) k (x) t k k! .
📌 3 Key Takeaways
🎯
Objective & Hypothesis:In the note, by virtue of the Fa\`a di Bruno formula and two identities for the Bell polynomials of the second kind, the authors derive three relations among the Bell polynomials, central factorial numbers of the second kind, and central Bell polynomials.
📊
Key Findings & Data:The Bell numbers Bk for k ≥ 0 can be generated [4, 7, 12] byee t−1 =X∞ k=0Bktk k!= 1 + t + t 2 +5 6 t 3 + 5 8 t 4 + 13 30 t 5 + 203 720 t 6 + 877 5040 t 7 + · · · As a generalization of the Bell numbers Bk for k ≥ 0, the Bell polynomials Tk(x) for k ≥ 0 can be generated [8– 10, 15, 17] by e x(e t−1) = X∞ k=0 Tk(x) t k k! = 1 + xt + 1 2 x(x + 1)t 2 + 1 6 x x 2 + 3x + 1 t 3 + 1 24 x x 3 + 6x 2 + 7x + 1 t 4 + 1 120 x x 4 + 10x 3 + 25x 2 + 15x + 1 t 5 + · · · (1.1) The polynomials Tk(x) for k ≥ 0 are also called [11, 18] the Touchard polynomials or the exponential polynomials.
💡
Conclusion & Impact:The central Bell polynomials B (c) k (x) for k ≥ 0 can be generated [5] by exp 2x sinh t 2 = X∞ k=0 B (c) k (x) t k k! .
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Bell polynomial · central factorial number of the second kind · central Bell polynomial
Abstract
In the note, by virtue of the Fa\`a di Bruno formula and two identities for the Bell polynomials of the second kind, the authors derive three relations among the Bell polynomials, central factorial numbers of the second kind, and central Bell polynomials. The Bell numbers Bk for k ≥ 0 can be generated [4, 7, 12] byee t−1 =X∞ k=0Bktk k!= 1 + t + t 2 +5 6 t 3 + 5 8 t 4 + 13 30 t 5 + 203 720 t 6 + 877 5040 t 7 + · · · As a generalization of the Bell numbers Bk for k ≥ 0, the Bell polynomials Tk(x) for k ≥ 0 can be generated [8– 10, 15, 17] by e x(e t−1) = X∞ k=0 Tk(x) t k k! = 1 + xt + 1 2 x(x + 1)t 2 + 1 6 x x 2 + 3x + 1 t 3 + 1 24 x x 3 + 6x 2 + 7x + 1 t 4 + 1 120 x x 4 + 10x 3 + 25x 2 + 15x + 1 t 5 + · · · (1.1) The polynomials Tk(x) for k ≥ 0 are also called [11, 18] the Touchard polynomials or the exponential polynomials. It is clear that Tk(1) = Bk. The central factorial numbers of the second kind T(n, k) for n ≥ k ≥ 0 can be generated [1, 6] by 1 k! 2 sinh t 2 k = X∞ n=k T(n, k) t n n! , where sinh t = e t − e −t 2 (1.2) is the hyperbolic sine function. The central Bell polynomials B (c) k (x) for k ≥ 0 can be generated [5] by exp 2x sinh t 2 = X∞ k=0 B (c) k (x) t k k! .
QI, F., GUO, B. (2019). Relations among Bell polynomials, central factorial numbers, and central Bell polynomials. Mathematical Sciences and Applications E-Notes, Cilt: 7 - Sayı: 2, 191–194. https://ajindex.com/articles/283477
\begin{thebibliography}{99} \bibitem{BSSV-NFAO-1989}P. L. Butzer, M. Schmidt, E. L. Stark, and L. Vogt, \emph{Central factorial numbers
their main properties and some applications}, Numer. Funct. Anal. Optim. \textbf{10} (1989), no.~5-6, 419\nobreakdash--488
Available online at \url{https://doi.org/10.1080/01630568908816313}. \bibitem{Charalambides-book-2002}C. A. Charalambides, \textit{Enumerative Combinatorics}, CRC Press Series on Discrete Mathematics and its Applications. Chapman \& Hall/CRC, Boca Raton, FL, 2002. \bibitem{Comtet-Combinatorics-74}L. Comtet, \textit{Advanced Combinatorics: The Art of Finite and Infinite Expansions}, Revised and Enlarged Edition, D. Reidel Publishing Co., 1974
Available online at \url{https://doi.org/10.1007/978-94-010-2196-8}. \bibitem{Bell-Stirling-HyperGeom.tex}B.-N. Guo and F. Qi, \textit{An explicit formula for Bell numbers in terms of Stirling numbers and hypergeometric functions}, Glob. J. Math. Anal. \textbf{2} (2014), no.~4, 243\nobreakdash--248
Available online at \url{http://dx.doi.org/10.14419/gjma.v2i4.3310}. \bibitem{Kim-Jang-Dolgy-Kim-ASCM-2019}D. S. Kim, G.-W. Jang, D. V. Dolgy, and T. Kim, \textit{An expression for central Bell polynomials}, Adv. Stud. Contemp. Math. \textbf{29} (2019), no.~2, 257\nobreakdash--262
Available online at \url{http://dx.doi.org/10.17777/ascm2019.29.2.257}. \bibitem{Kim2Russ-2019}T. Kim and D. S. Kim, \textit{A note on central Bell numbers and polynomials}, to appear in Russ. J. Math. Phys. (2019), in press. \bibitem{Merca-Period-2016}M. Merca, \emph{Connections between central factorial numbers and Bernoulli polynomials}, Period. Math. Hungar. \textbf{73} (2016), no.~2, 259\nobreakdash--264
Available online at \url{https://doi.org/10.1007/s10998-016-0140-5}. \bibitem{Bell-Stirling-Lah-simp.tex}F. Qi, \textit{An explicit formula for the Bell numbers in terms of the Lah and Stirling numbers}, Mediterr. J. Math. \textbf{13} (2016), no.~5, 2795\nobreakdash--2800
Available online at \url{https://doi.org/10.1007/s00009-015-0655-7}. \bibitem{Log-Poly-Prop-IIS.tex}F. Qi, \textit{Integral representations for multivariate logarithmic polynomials}, J. Comput. Appl. Math. \textbf{336} (2018), 54\nobreakdash--62
Available online at \url{https://doi.org/10.1016/j.cam.2017.11.047}. \bibitem{Log-Poly-Prop.tex}F. Qi, \textit{On multivariate logarithmic polynomials and their properties}, Indag. Math. (N.S.) \textbf{29} (2018), no.~5, 1179\nobreakdash--1192