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Research Article

Open Access Peer-reviewed

Feng Qi, Guo-Sheng Wu, Bai-Ni Guo^{ }

Received March 13, 2019; Revised April 19, 2019; Accepted April 24, 2019

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.

**2010 Mathematics Subject Classi****fi*** cation*. Primary 11B83; Secondary 11B75, 33B10.

In mathematics, a closed formula is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, four arithmetic operations, and elementary functions, but usually no limit.

The Bell polynomials of the second kind, denoted by for are defined ^{ 1, 2, 3, 4, 5, 6, 7} by

The central factorial numbers of the second kind for can be generated ^{ 8, 9, 10} by

(1.1) |

The central factorial numbers of the first kind for can be generated ^{ 8, 9} by

(1.2) |

In [^{ 8}, Proposition 2.4, (xii)], the authors established the closed formula

(1.3) |

In this short note, by virtue of several formulas and identities for special values of the Bell polynomials of the second kind , we will provide an alternative proof of the closed formula (1.3).

For alternatively proving the closed formula (1.3), we need the following lemmas.

**Lemma 2.1** (^{ 1, 2, 11, 12, 13, 14}). T*he Faà di Bruno formula** can be described in** **terms of** ** **by*

(2.1) |

*For **, the Bell polynomials of the second kind ** satisfy the* identity

(2.2) |

*is valid, where *

**Lemma 2.2** ^{ 10, 11, 12, 14, 15, 16, 17, 18}. *For *, *the Bell polynomials of the second kind ** satisfy the closed formula*

(2.3) |

*where ** is regarded as* *and ** denote the associate Stirling numbers of the second kind or weighted Stirling numbers which can be generated by*

The closed formula (1.3) can be rewritten in terms of the associate Stirling numbers of the second kind or weighted Stirling numbers as follows.

**Theorem 3.1.*** **For ** the central factorial numbers of the second kind ** satisfy*

(3.1) |

*Proof.* The equation (1.1) implies that

(3.2) |

Let . By virtue of the Faà* *di Bruno formula (2.1), we obtain

where the quantity

is called ^{ 2, 19} the falling factorial of *x*.

Since

for as , it follows that

where we used the identity (2.2) and the closed formula (2.3). Consequently, the formula (3.1) follows immediately. The proof of Theorem 3.1 is complete.

In this section, we pose two open problems.

For alternatively and similarly finding a closed formula for the central factorial numbers of the first kind generated in (1.2), we need to solve the following open problem.

**Open Problem 4.1.** *Can one **fi**nd a closed formula of the Bell polynomials of the** **second kind*

*for *?

For and , the falling factorial is defined by

(4.1) |

In ^{ 12, 20} and closely related references therein, the following conclusions were obtained.

**Theorem 4.1** ([^{ 20}, Theorem 3.1] and [^{ 12}, Section 1.4]). *For *, *we have*

(4.2) |

**Theorem 4.2** ([^{ 20}, Theorem 3.2] and [^{ 12}, Section 1.5]). *For *, *we have*

(4.3) |

*where the double factorial of negative odd integers ** for ** is defined by*

The formula (4.3) has been applied in [^{ 21}, Theorem 1.1].

Motivated by the above conclusions, one can naturally pose the following open problem.

**Open Problem 4.2.**** ***For ** and **, can one find an explicit, elementary, simple, and general formula of the type as in** *(4.2) *and** *(4.3) * for the finite sum*

*In particular, how about special cases ** for * *in** *(4.4)?

[1] | C. A. Charalambides, Enumerative Combinatorics, CRC Press Series on Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, FL, 2002. | ||

In article | |||

[2] | L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., 1974. | ||

In article | |||

[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. | ||

In article | |||

[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. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[9] | M. Merca, Connections between central factorial numbers and Bernoulli polynomials, Period. Math. Hungar. 73 (2016), no. 2, 259-264. | ||

In article | View Article | ||

[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. | ||

In article | |||

[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. | ||

In article | View Article | ||

[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. | ||

In article | |||

[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. | ||

In article | View Article | ||

[14] | C.-F. Wei and F. Qi, Several closed expressions for the Euler numbers, J. Inequal. Appl. 2015, 2015: 219, 8 pages. | ||

In article | View Article | ||

[15] | A. Z. Broder, The r-Stirling numbers, Discrete Math. 49 (1984), no. 3, 241-259. | ||

In article | View Article | ||

[16] | L. Carlitz, Weighted Stirling numbers of the first and second kind, I, Fibonacci Quart. 18 (1980), no. 2, 147-162. | ||

In article | |||

[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. | ||

In article | |||

[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. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[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. | ||

In article | |||

[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. | ||

In article | |||

Published with license by Science and Education Publishing, Copyright © 2019 Feng Qi, Guo-Sheng Wu and Bai-Ni Guo

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Feng Qi, Guo-Sheng Wu, 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*. Vol. 7, No. 2, 2019, pp 56-58. http://pubs.sciepub.com/tjant/7/2/5

Qi, Feng, 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* 7.2 (2019): 56-58.

Qi, F. , Wu, G. , & Guo, B. (2019). An Alternative Proof of a Closed Formula for Central Factorial Numbers of the Second Kind. *Turkish Journal of Analysis and Number Theory*, *7*(2), 56-58.

Qi, Feng, 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* 7, no. 2 (2019): 56-58.

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[1] | C. A. Charalambides, Enumerative Combinatorics, CRC Press Series on Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, FL, 2002. | ||

In article | |||

[2] | L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., 1974. | ||

In article | |||

[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. | ||

In article | |||

[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. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[9] | M. Merca, Connections between central factorial numbers and Bernoulli polynomials, Period. Math. Hungar. 73 (2016), no. 2, 259-264. | ||

In article | View Article | ||

[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. | ||

In article | |||

[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. | ||

In article | View Article | ||

[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. | ||

In article | |||

[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. | ||

In article | View Article | ||

[14] | C.-F. Wei and F. Qi, Several closed expressions for the Euler numbers, J. Inequal. Appl. 2015, 2015: 219, 8 pages. | ||

In article | View Article | ||

[15] | A. Z. Broder, The r-Stirling numbers, Discrete Math. 49 (1984), no. 3, 241-259. | ||

In article | View Article | ||

[16] | L. Carlitz, Weighted Stirling numbers of the first and second kind, I, Fibonacci Quart. 18 (1980), no. 2, 147-162. | ||

In article | |||

[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. | ||

In article | |||

[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. | ||

In article | View Article | ||

[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. | ||

In article | View Article | ||

[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. | ||

In article | |||

[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. | ||

In article | |||