A Qi formula for translated \(r\)-Dowling numbers
Volume 20, Issue 2, pp 88--100
http://dx.doi.org/10.22436/jmcs.020.02.02
Publication Date: October 19, 2019
Submission Date: June 28, 2019
Revision Date: September 07, 2019
Accteptance Date: September 10, 2019
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Authors
Roberto B. Corcino
- Research Institute for Computational Mathematics and Physics, Cebu Normal University, Osmena Boulevard, Cebu City, Philippines.
Cristina B. Corcino
- Research Institute for Computational Mathematics and Physics, Cebu Normal University, Osmena Boulevard, Cebu City, Philippines.
Jeneveb T. Malusay
- Research Institute for Computational Mathematics and Physics, Cebu Normal University, Osmena Boulevard, Cebu City, Philippines.
Abstract
Another form of an explicit formula for translated \(r\)-Dowling numbers is derived using Faa di Bruno's formula and certain identity of Bell polynomials of the second kind. This formula is expressed in terms of the translated \(r\)-Whitney numbers of the second kind and the ordinary Lah numbers, which is analogous to Qi formula. As a consequence, a relation between translated \(r\)-Dowling numbers and the sums of row entries of the product of two matrices containing the translated \(r\)-Whitney numbers of the second kind and the ordinary Lah numbers is established.
Share and Cite
ISRP Style
Roberto B. Corcino, Cristina B. Corcino, Jeneveb T. Malusay, A Qi formula for translated \(r\)-Dowling numbers, Journal of Mathematics and Computer Science, 20 (2020), no. 2, 88--100
AMA Style
Corcino Roberto B., Corcino Cristina B., Malusay Jeneveb T., A Qi formula for translated \(r\)-Dowling numbers. J Math Comput SCI-JM. (2020); 20(2):88--100
Chicago/Turabian Style
Corcino, Roberto B., Corcino, Cristina B., Malusay, Jeneveb T.. "A Qi formula for translated \(r\)-Dowling numbers." Journal of Mathematics and Computer Science, 20, no. 2 (2020): 88--100
Keywords
- Qi formula
- Translated \(r\)-Dowling numbers
- Bell polynomials
- Lah numbers
- translated \(r\)-Whitney numbers
- Faa di Bruno's formula
- \(r\)-Whitney-Lah numbers
MSC
References
-
[1]
I. Area, E. Godoy, A. Ronveaux, A. Zarzo, Classical Discrete Orthogonal Polynomials, Lah Numbers, and Involutory Matrices, Appl. Math. Lett., 16 (2003), 383--387
-
[2]
H. Belbachir, I. E. Bousbaa, Translated Whitney and $r$-Whitney Numbers: A Combinatorial Approach, J. Integer Seq., 16 (2013), 7 pages
-
[3]
M. Benoumhani, On Whitney Numbers of Dowling Lattices, Discrete Math., 159 (1996), 13--33
-
[4]
K. N. Boyadzhiev, Lah Numbers, Laguerre Polynomials of Order Negative One, and the $n$th Derivative of $\exp(1/x)$, Acta Univ. Sapientiae Math., 8 (2016), 22--31
-
[5]
A. Z. Broder, The $r$-Stirling Numbers, Discrete Math., 49 (1984), 241--259
-
[6]
C. C. Chen, K.-M. Koh, Principles and Techniques in Combinatorics, World Scientific Publ. Co., Singapore (1992)
-
[7]
G.-S. Cheon, J.-H. Jung, $r$-Whitney numbers of Dowling Lattices, Discrete Math., 312 (2012), 2337--2348
-
[8]
L. Comtet, Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht (1974)
-
[9]
S. Daboul, J. Mangaldan, M. Z. Spivey, P. J. Taylor, The Lah Numbers and the $n$th Derivative of $e^\frac{1}{x}$, Math. Mag., 86 (2013), 39--47
-
[10]
T. A. Dowling, A Class of Geometric Lattices Based on Finite Groups, J. Combinatorial Theory Ser. B, 14 (1973), 61--86
-
[11]
B.-N. Guo, F. Qi, Six Proofs for an Identity of the Lah Numbers, Online J. Anal. Comb., 10 (2015), 5 pages
-
[12]
E. Gyimesi, G. Nyul, A comprehensive study of $r$-Dowling polynomials, Aequationes Math., 92 (2018), 515--527
-
[13]
E. Gyimesi, G. Nyul, New combinatorial interpretations of $r$-Whitney and r-Whitney-Lah numbers, Discrete Appl. Math., 255 (2019), 222--233
-
[14]
L. C. Hsu, P. J.-S. Shiue, A unified approach to generalized Stirling numbers, Adv. in Appl. Math., 20 (1998), 366--384
-
[15]
I. Lah, IEine neue Art von Zahlen, ihre Eigenschaften und Anwendung in der mathematischen Statistik, (German) Mitteilungsbl. Math. Statist., 7 (1955), 203--212
-
[16]
P. Lancaster, M. Tismenetsky, The Theory of Matrices, Academic Press, Orlando (1985)
-
[17]
M. Merca, A New Connection between $r$-Whitney numbers and Bernuolli Polynomials, Integral Transforms Spec. Funct., 25 (2014), 937--942
-
[18]
I. Mező, New Properties of $r$-Stirling Series, Acta Math. Hungar., 119 (2008), 341--358
-
[19]
I. Mező, On the Maximum of $r$-Stirling numbers, Adv. in Appl. Math., 41 (2008), 293--306
-
[20]
I. Mező, A New Formula for the Bernoulli Polynomials, Results Math., 58 (2010), 329--335
-
[21]
I. Mező, The $r$-Bell Numbers, J. Integer Seq., 14 (2011), 14 pages
-
[22]
F. Qi, An Explicit Formula for the Bell Numbers in terms of Lah and Stirling Numbers, Mediterr. J. Math., 13 (2016), 2795--2800
-
[23]
J. Riordan, Introduction to Combinatorial Analysis, John Wiley & Sons, New York (1958)
-
[24]
C. C. Wagner, Generalized Stirling and Lah numbers, Discrete Math., 160 (1996), 199--218
-
[25]
X.-J. Zhang, F. Qi, W.-H. Li, Properties of three functions relating to the exponential function and the existence of partitions of unity, Int. J. Open Probl. Comput. Sci. Math., 5 (2012), 122--127
