A \(q\)-analogue of \(r\)-Whitney numbers of the second kind and its Hankel transform
Volume 21, Issue 3, pp 258--272
http://dx.doi.org/10.22436/jmcs.021.03.08
Publication Date: April 29, 2020
Submission Date: January 28, 2020
Revision Date: February 21, 2020
Accteptance Date: March 18, 2020
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Authors
Roberto B. Corcino
- Research Institute for Computational Mathematics and Physics, Cebu Normal University, Osmeña Boulevard, Cebu City, Philippines.
Jay M. Ontolan
- Research Institute for Computational Mathematics and Physics, Cebu Normal University, Osmeña Boulevard, Cebu City, Philippines.
Jennifer Cañete
- Research Institute for Computational Mathematics and Physics, Cebu Normal University, Osmeña Boulevard, Cebu City, Philippines.
Mary Joy R. Latayada
- Mathematics Department, Caraga State University, Butuan City, Philippines.
Abstract
A \(q\)-analogue of \(r\)-Whitney numbers of the second kind, denoted by \(W_{m,r}[n,k]_q\), is defined by means of a triangular recurrence relation. In this paper, several fundamental properties for the said \(q\)-analogue are established including other forms of recurrence relations, explicit formulas and generating functions. Moreover, a kind of Hankel transform for \(W_{m,r}[n,k]_q\) is obtained.
Share and Cite
ISRP Style
Roberto B. Corcino, Jay M. Ontolan, Jennifer Cañete, Mary Joy R. Latayada, A \(q\)-analogue of \(r\)-Whitney numbers of the second kind and its Hankel transform, Journal of Mathematics and Computer Science, 21 (2020), no. 3, 258--272
AMA Style
Corcino Roberto B., Ontolan Jay M., Cañete Jennifer, Latayada Mary Joy R., A \(q\)-analogue of \(r\)-Whitney numbers of the second kind and its Hankel transform. J Math Comput SCI-JM. (2020); 21(3):258--272
Chicago/Turabian Style
Corcino, Roberto B., Ontolan, Jay M., Cañete, Jennifer, Latayada, Mary Joy R.. "A \(q\)-analogue of \(r\)-Whitney numbers of the second kind and its Hankel transform." Journal of Mathematics and Computer Science, 21, no. 3 (2020): 258--272
Keywords
- \(r\)-Whitney numbers
- \(r\)-Dowling numbers
- generating function
- \(q\)-exponential function
- symmetric function
- Hankel transform
MSC
References
-
[1]
M. Aigner, A Characterization of the Bell Numbers, Discrete Math., 205 (1999), 207--210
-
[2]
W. M. Bent-Usman, A. M. Dibagulun, M. M. Mangontarum, C. B. Montero, An Alternative $q$-Analogue of the Rucinski-Voigt Numbers, Commun. Korean Math. Soc., 33 (2018), 1055--1073
-
[3]
L. Carlitz, $q$-Bernoulli numbers and polynomials, Duke Math. J., 15 (1948), 987--1000
-
[4]
G.-S. Cheon, J.-H. Jung, $r$-Whitney numbers of Dowling Lattices, Discrete Math., 312 (2012), 2337--2348
-
[5]
L. Comtet, Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht (1974)
-
[6]
R. B. Corcino, The $(r,\beta)$-Stirling numbers, Mindanao Forum, 14 (1999), 91--99
-
[7]
R. B. Corcino, C. B. Corcino, The Hankel Transform of Generalized Bell Numbers and Its $q$-Analogue, Util. Math., 89 (2012), 297--309
-
[8]
R. B. Corcino, C. B. Corcino, R. Aldema, Asymptotic Normality of the $(r,\beta)$-Stirling Numbers, Ars Combin., 81 (2006), 81--96
-
[9]
C. B. Corcino, R. B. Corcino, J. M. Ontolan, C. M. Perez-Fernandez, E. R. Cantallopez, The Hankel Transform of $q$-Noncentral Bell Numbers, Int. J. Math. Math. Sci., 2015 (2015), 10 pages
-
[10]
R. B. Corcino, M. J. R. Latayada, M. A. Vega, The Hankel Transform of $(q,r)$-Dowling Numbers, Eur. J. Pure Appl. Math., 12 (2019), 279--293
-
[11]
R. B. Corcino, C. B. Montero, A $q$-Analogue of Rucinski-Voigt Numbers, ISRN Discrete Math., 2012 (2012), 18 pages
-
[12]
A. de Medicis, P. Leroux, Generalized Stirling Numbers, Convolution Formulae and $p,q$-Analogues, Can. J. Math., 47 (1995), 474--499
-
[13]
R. Ehrenborg, The Hankel Determinant of exponential Polynomials, Amer. Math. Monthly, 107 (2000), 557--560
-
[14]
T. Ernst, A Method for $q$-calculus, J. Nonlinear Math. Phys., 10 (2003), 487--525
-
[15]
H. Exton, $q$-Hypergeometric Functions and Applications, Halstead Press, New York (1983)
-
[16]
M. Garcia Armas, B. A. Seturaman, A note on the Hankel transform of the central binomial coefficients, J. Integer Seq., 11 (2008), 9 pages
-
[17]
H. W. Gould, The $q$-Stirling Number of the First and Second Kinds, Duke Math. J., 28 (1968), 281--289
-
[18]
J. Katriel, M. Kibler, Normal ordering for deformed boson operators and operator-valued deformed Stirling numbers, J. Phys. A, 25 (1992), 2683--2691
-
[19]
M.-S. Kim, J.-W. Son, A Note on $q$-Difference Operators, Commun. Korean Math. Soc., 17 (2002), 423--430
-
[20]
M. Koutras, Non-Central Stirling Numbers and Some Applications, Discrete Math., 42 (1982), 73--89
-
[21]
J. W. Layman, The Hankel transform and some of its properties, J. Integer Seq., 4 (2001), 11 pages
-
[22]
M. Mangontarum, J. Katriel, On $q$-Boson Operators and $q$-Analogues of the $r$-Whitney and $r$-Dowling Numbers, J. Integer Seq., 18 (2015), 23 pages
-
[23]
T. Mansour, M. Schork, Commutation Relations, Normal Ordering, and Stirling Numbers, CRC Press, Boca Raton (2016)
-
[24]
I. Mező, A New Formula for the Bernoulli Polynomials, Results Math., 58 (2010), 329--335
-
[25]
I. Mező, The $r$--Bell numbers, J. Integer Seq., 14 (2011), 14 pages
-
[26]
C. Sun, N. A. Sinitsyn, Landau-Zener extension of the Tavis-Cummings model: Structure of the solution, Phys. Rev. A, 94 (2016), 9 pages
