cA \(q\)-analogue of generalized translated Whitney numbers: Cigler's approach
Volume 24, Issue 1, pp 82--96
http://dx.doi.org/10.22436/jmcs.024.01.08
Publication Date: January 18, 2021
Submission Date: August 26, 2020
Revision Date: December 03, 2020
Accteptance Date: December 12, 2020
-
1254
Downloads
-
3127
Views
Authors
Roberto B. Corcino
- Research Institute for Computational Mathematics and Physics, Cebu Normal University, Cebu City 6000, Philippines.
Jezer C. Fernandez
- Department of Mathematics, Mindanao State University, Marawi City 9700, Philippines.
Mary Ann Ritzell P. Vega
- Department of Mathematics and Statistics, College of Science and Mathematics, Mindanao State University-Iligan Institute of Technology, Iligan City 9200, Philippines.
Abstract
Using a certain combinatorial interpretation in terms of set partition, a \(q\)-analogue of generalized translated Whitney numbers of the second kind is defined in this paper. Some properties such as the recurrence relation, explicit formula, and certain symmetric formula are obtained. Moreover, a \(q\)-analogue of generalized translated Whitney numbers of the first kind is introduced to obtain a \(q\)-analogue of the orthogonality and inverse relations of the two kinds of generalized translated Whitney numbers.
Share and Cite
ISRP Style
Roberto B. Corcino, Jezer C. Fernandez, Mary Ann Ritzell P. Vega, cA \(q\)-analogue of generalized translated Whitney numbers: Cigler's approach, Journal of Mathematics and Computer Science, 24 (2022), no. 1, 82--96
AMA Style
Corcino Roberto B., Fernandez Jezer C., Vega Mary Ann Ritzell P., cA \(q\)-analogue of generalized translated Whitney numbers: Cigler's approach. J Math Comput SCI-JM. (2022); 24(1):82--96
Chicago/Turabian Style
Corcino, Roberto B., Fernandez, Jezer C., Vega, Mary Ann Ritzell P.. "cA \(q\)-analogue of generalized translated Whitney numbers: Cigler's approach." Journal of Mathematics and Computer Science, 24, no. 1 (2022): 82--96
Keywords
- Stirling numbers
- \(r\)-Stirling numbers
- translated Whitney numbers
- \(q\)-binomial coefficients
- \(q\)-factorial
MSC
References
-
[1]
H. Belbachir, I. E. Bousbaa, Translated Whitney and r-Whitney numbers: a combinatorial approach, J. Integer Seq., 16 (2013), 7 pages
-
[2]
M. Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math., 159 (1996), 13--33
-
[3]
A. Z. Broder, The r-Stirling numbers, Discrete Math., 49 (1984), 241--259
-
[4]
L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J., 15 (1948), 987--1000
-
[5]
J. Cigler, A new q-analog of Stirling numbers, Osterreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, 201 (1992), 97--109
-
[6]
J. Cigler, Operatormethoden für $q$-Identitäten, Monatsh. Math., 88 (1979), 87--105
-
[7]
L. Comtet, Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht (1974)
-
[8]
C. B. Corcino, An asymptotic formula for the r-Bell numbers, Matimyás Mat., 24 (2001), 9--18
-
[9]
R. B. Corcino, C. B. Corcino, R. Aldema, Asymptotic normality of the $(r,\beta)$-Stirling numbers, Ars Combin., 81 (2006), 81--96
-
[10]
R. B. Corcino, J. C. Fernandez, A Combinatorial Approach for q-Analogue of r-Stirling Numbers, J. adv. math. comput. sci., 4 (2014), 1268--1279
-
[11]
R. B. Corcino, C. B. Corcino, The Hankel transform of generalized Bell numbers and its q-analogue, Util. Math., 89 (2012), 297--309
-
[12]
R. B. Corcino, M. J. Latayada, MAR. Vega, Hankel Transform of (q, r)-Dowling Numbers, Eur. J. Pure Appl. Math., 12 (2019), 279--293
-
[13]
R. B. Corcino, J. T. Malusay, J. A. Cillar, G. J. Rama, O. V. Silang, I. M. Tacoloy, Analogies of the Qi formula for some Dowling type numbers, Util. Math., 111 (2019), 3--26
-
[14]
R. B. Corcino, C. B. Montero, A q-Analogue of Rucinski-Voigt Numbers, ISRN Discrete Math., 2012 (2012), 18 pages
-
[15]
R. B. Corcino, J. M. Ontolan, J. Canete, M. R. Latayad, A q-Analogue of r-Whitney Numbers of the Second Kind and Its Hankel Transform, J. Math. Comput. Sci., 21 (2020), 258--272
-
[16]
] R. B. Corcino, J. M. Ontolan, G. J. Rama, ,, 12 (2019), 1676–1688. 4, Hankel transform of the second form (q, r)-Dowling numbers, Eur. J. Pure Appl. Math., 12 (2019), 1676--1688
-
[17]
T. A. Dowling, A Class of Geometric Lattices Based on Finite Groups, J. Combinatorial Theory Ser. B, 14 (1973), 61--86
-
[18]
H. W. Gould, The q-Stirling numbers of first and second kinds, Duke Math. J., 28 (1961), 281--289
-
[19]
L. C. Hsu, P. J.-S. Shiue, A unified approach to generalized Stirling numbers, Adv. in Appl. Math., 20 (1998), 366--384
-
[20]
F. H. Jackson, Certain q-identities, Quart. J. Math., 12 (1941), 167--172
-
[21]
T. Mansour, Combinatorics of Set Partitions, Discrete Mathematics and Its Applications Series, CRC Press, Boca Raton, FL (2012)
-
[22]
T. Mansour, M. Schork, M. Shattuck, On a new family of generalized Stirling and Bell numbers, Electron. J. Combin., 18 (2011), 33 pages
-
[23]
T. Mansour, M. Schork, M. Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Appl. Math. Lett., 25 (2012), 1767--1771
-
[24]
T. Mansour, M. Schork, M. Shattuck, The generalized Stirling and Bell numbers revisited, J. Integer Seq., 15 (2012), 47 pages
-
[25]
I. Mező, On the Maximum of r-Stirling numbers, Adv. in Appl. Math., 41 (2008), 293--306
-
[26]
I. Mező, New properties of r-Stirling series, Acta Math. Hungar., 119 (2008), 341--358
-
[27]
I. Mező, The r-Bell numbers, J. Integer Seq., 14 (2011), 14 pages
-
[28]
M. Mihoubi, M. S. Maamra, The (r1 , . . . , rp)-Stirling Numbers of the Second Kind, Integers, 12 (2012), 787--1089
-
[29]
C. G. Wagner, Partition statistics and $q$-Bell numbers $(q=-1)$, J. Integer Seq., 7 (2004), 12 pages
