Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

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

Feng Qi, Guo-Sheng Wu, Bai-Ni Guo*

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

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

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

Citation: 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, 7 (2). 56-58. January 2019. doi:10.12691/tjant-7-2-5

2010 Mathematics Subject Classification. Primary 11B83; Secondary 11B75, 33B10.

1. Introduction

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

[picture] | (1.1)

The central factorial numbers of the first kind for can be generated [8,9] by

[picture] | (1.2)

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

[picture] | (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).

2. Lemmas

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

Lemma 2.1 ([1,2,11,12,13,14]). The Faà di Bruno formula can be described in terms of by

[picture] | (2.1)

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

[picture] | (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

[picture] | (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

3. An Alternative Proof of the Closed Formula (1.3)

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

[picture] | (3.1)

Proof. The equation (1.1) implies that

[picture] | (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.

4. Two Open Problems

In this section, we pose two open problems.

4.1. First Open Problem

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 find a closed formula of the Bell polynomials of the second kind

for ?

4.2. Second Open Problem.

For and , the falling factorial is defined by

[picture] | (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

[picture] | (4.2)

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

[picture] | (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)?

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.