Turkish Journal of Analysis and Number Theory
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Turkish Journal of Analysis and Number Theory. 2019, 7(2), 56-58
DOI: 10.12691/tjant-7-2-5
Open AccessArticle

An Alternative Proof of a Closed Formula for Central Factorial Numbers of the Second Kind

Feng Qi1, 2, Guo-Sheng Wu3 and Bai-Ni Guo4,

1College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, 028043, Inner Mongolia, China

2School of Mathematical Sciences, Tianjin Polytechnic, University, Tianjin 300387, China

3School of Computer Science, Sichuan Technology and Business University, Chengdu 611745, Sichuan, China

4School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo, 454010, Henan, China

Pub. Date: April 24, 2019

Cite this paper:
Feng Qi, Guo-Sheng Wu and Bai-Ni Guo. An Alternative Proof of a Closed Formula for Central Factorial Numbers of the Second Kind. Turkish Journal of Analysis and Number Theory. 2019; 7(2):56-58. doi: 10.12691/tjant-7-2-5

Abstract

In the short note, by virtue of several formulas and identities for special values of the Bell polynomials of the second kind, the authors provide an alternative proof of a closed formula for central factorial numbers of the second kind. Moreover, the authors pose two open problems on closed form of a special Bell polynomials of the second kind and on closed form of a finite sum involving falling factorials.

Keywords:
alternative proof closed formula central factorial number of the second kind Bell polynomial of the second kind finite sum falling factorial open problem

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References:

[1]  C. A. Charalambides, Enumerative Combinatorics, CRC Press Series on Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, FL, 2002.
 
[2]  L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., 1974.
 
[3]  F. Qi, V. Čerňanová, and Y. S. Semenov, 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. 81 (2019), no. 1, 123-136.
 
[4]  F. Qi, D. Lim, and B.-N. Guo, Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113 (2019), no. 1, 1-9.
 
[5]  F. Qi, D. Lim, and Y.-H. Yao, Notes on two kinds of special values for the Bell polynomials of the second kind, Miskolc Math. Notes 20 (2019), no. 1, 465-474.
 
[6]  F. Qi and A.-Q. Liu, Alternative proofs of some formulas for two tridiagonal determinants, Acta Univ. Sapientiae Math. 10 (2018), no. 2, 287-297.
 
[7]  F. Qi, D.-W. Niu, and B.-N. Guo, Some identities for a sequence of unnamed polynomials connected with the Bell polynomials, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 113 (2019), no. 2, 557-567.
 
[8]  P. L. Butzer, M. Schmidt, E. L. Stark, and L. Vogt, Central factorial numbers; their main properties and some applications, Numer. Funct. Anal. Optim. 10 (1989), no. 5-6, 419-488.
 
[9]  M. Merca, Connections between central factorial numbers and Bernoulli polynomials, Period. Math. Hungar. 73 (2016), no. 2, 259-264.
 
[10]  F. Qi and B.-N. Guo, Relations among Bell polynomials, central factorial numbers, and central Bell polynomials, Math. Sci. Appl. E-Notes 7 (2019), no. 2, 191-194.
 
[11]  F. Qi and B.-N. Guo, Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials, Mediterr. J. Math. 14 (2017), no. 3, Article 140, 14 pages.
 
[12]  F. Qi, D.-W. Niu, D. Lim, and Y.-H. Yao, Special values of the Bell polynomials of the second kind for some sequences and functions, HAL archives (2018), available online at https://hal.archives-ouvertes.fr/hal-01766566.
 
[13]  F. Qi and M.-M. Zheng, Explicit expressions for a family of the Bell polynomials and applications, Appl. Math. Comput. 258 (2015), 597-607.
 
[14]  C.-F. Wei and F. Qi, Several closed expressions for the Euler numbers, J. Inequal. Appl. 2015, 2015: 219, 8 pages.
 
[15]  A. Z. Broder, The r-Stirling numbers, Discrete Math. 49 (1984), no. 3, 241-259.
 
[16]  L. Carlitz, Weighted Stirling numbers of the first and second kind, I, Fibonacci Quart. 18 (1980), no. 2, 147-162.
 
[17]  M. Griffiths and I. Mezö, A generalization of Stirling numbers of the second kind via a special multiset, J. Integer Seq. 13 (2010), no. 2, Article 10.2.5, 23 pp.
 
[18]  B.-N. Guo, I. Mezö, and F. Qi, An explicit formula for the Bernoulli polynomials in terms of the r-Stirling numbers of the second kind, Rocky Mountain J. Math. 46 (2016), no. 6, 1919-1923.
 
[19]  F. Qi, X.-T. Shi, and F.-F. Liu, Several identities involving the falling and rising factorials and the Cauchy, Lah, and Stirling numbers, Acta Univ. Sapientiae Math. 8 (2016), no. 2, 282-297.
 
[20]  F. Qi, D.-W. Niu, D. Lim, and B.-N. Guo, Closed formulas and identities for the Bell polynomials and falling factorials, Contrib. Discrete Math. 14 (2019), no. 2, in press.
 
[21]  F. Qi, Simplifying coefficients in differential equations related to generating functions of reverse Bessel and partially degenerate Bell polynomials, Bol. Soc. Paran. Mat. 39 (2021), no. 4, in press.