Relations among Bell polynomials, central factorial numbers, and central Bell polynomials

Year 2019, Volume: 7 Issue: 2, 191 - 194, 15.10.2019

Feng Qi , Bai-Ni Guo

https://doi.org/10.36753/mathenot.566448

Cited By: 4

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! . 

Keywords

Bell polynomial, central factorial number of the second kind, central Bell polynomial, Bell polynomial of the second kind, Faà di Bruno formula

References

• \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; Available online at \url{https://doi.org/10.1016/j.indag.2018.04.002}.
• \bibitem{K2Jang-ascm17.tex}F. Qi, \textit{Simplifying coefficients in differential equations related to generating functions of reverse Bessel and partially degenerate Bell polynomials}, Bol. Soc. Paran. Mat. \textbf{39} (2021), no.~4, in press; Available online at \url{http://dx.doi.org/10.5269/bspm.41758}.
• \bibitem{MIA-6530.tex}F. Qi, \textit{Some inequalities and an application of exponential polynomials}, Math. Inequal. Appl. \textbf{22} (2019), in press; Available online at \url{https://doi.org/10.7153/mia-2019-22-??}.
• \bibitem{Bell-Num-Ineq.tex}F. Qi, \textit{Some inequalities for the Bell numbers}, Proc. Indian Acad. Sci. Math. Sci. \textbf{127} (2017), no.~4, 551\nobreakdash--564; Available online at \url{https://doi.org/10.1007/s12044-017-0355-2}.
• \bibitem{Det-Tri-Diag-S.tex}F. Qi, V. \v{C}er\v{n}anov\'a, and Y. S. Semenov, \textit{Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials}, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. \textbf{81} (2019), no.~1, 123\nobreakdash--136.
• \bibitem{Spec-Bell2Euler-S.tex}F. Qi and B.-N. Guo, \textit{Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials}, Mediterr. J. Math. \textbf{14} (2017), no.~3, Article~140, 14~pages; Available online at \url{https://doi.org/10.1007/s00009-017-0939-1}.
• \bibitem{Qi-Lim-Zhao-Bell.tex}F. Qi, D. Lim, and B.-N. Guo, \emph{Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations}, Rev. R. Acad. Cienc. Exactas F\'is. Nat. Ser. A Mat. RACSAM \textbf{113} (2019), no.~1, 1\nobreakdash--9; Available online at \url{https://doi.org/10.1007/s13398-017-0427-2}.
• \bibitem{Char-Bell-Qi-MNN.tex}F. Qi, D. Lim, and Y.-H. Yao, \textit{Notes on two kinds of special values for the Bell polynomials of the second kind}, Miskolc Math. Notes \textbf{20} (2019), no.~2, in press; Available online at \url{https://doi.org/10.18514/MMN.2019.2635}.
• \bibitem{QLG-RACSAM-Ext.tex}F. Qi, D.-W. Niu, and B.-N. Guo, \textit{Some identities for a sequence of unnamed polynomials connected with the Bell polynomials}, Rev. R. Acad. Cienc. Exactas F\'is. Nat. Ser. A Math. RACSAM \textbf{113} (2019), no.~2, 557\nobreakdash--567; Available online at \url{https://doi.org/10.1007/s13398-018-0494-z}.
• \bibitem{Bell-Poly-Gen-Boy.tex}F. Qi, D.-W. Niu, D. Lim, and B.-N. Guo, \emph{Some properties and an application of multivariate exponential polynomials}, HAL archives (2018), available online at \url{https://hal.archives-ouvertes.fr/hal-01745173}.
• \bibitem{Bell-value-elem-funct.tex}F. Qi, D.-W. Niu, D. Lim, and Y.-H. Yao, \textit{Special values of the Bell polynomials of the second kind for some sequences and functions}, HAL archives (2018), available online at \url{https://hal.archives-ouvertes.fr/hal-01766566}.
• \bibitem{Deriv-Arcs-Cos.tex}F. Qi and M.-M. Zheng, \textit{Explicit expressions for a family of the Bell polynomials and applications}, Appl. Math. Comput. \textbf{258} (2015), 597\nobreakdash--607; Available online at \url{https://doi.org/10.1016/j.amc.2015.02.027}.
• \bibitem{centr-bell-polyn.tex}F. Qi, G.-S. Wu, and B.-N. Guo, \textit{An alternative proof of a closed formula for central factorial numbers of the second kind}, submitted.
• \bibitem{Euler-No-3Sum.tex}C.-F. Wei and F. Qi, \textit{Several closed expressions for the Euler numbers}, J. Inequal. Appl. 2015, \textbf{2015}:219, 8~pages; Available online at \url{https://doi.org/10.1186/s13660-015-0738-9}.
• \end{thebibliography}