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http://dx.doi.org/10.4134/CKMS.c170379

AN ALTERNATIVE q-ANALOGUE OF THE RUCINSKI-VOIGT NUMBERS  

Bent-Usman, Wardah M. (Department of Mathematics Mindanao State University-Main Campus)
Dibagulun, Amerah M. (Department of Mathematics Mindanao State University-Main Campus)
Mangontarum, Mahid M. (Department of Mathematics Mindanao State University-Main Campus)
Montero, Charles B. (Department of Mathematics Mindanao State University-Main Campus)
Publication Information
Communications of the Korean Mathematical Society / v.33, no.4, 2018 , pp. 1055-1073 More about this Journal
Abstract
In this paper, we define an alternative q-analogue of the $Ruci{\acute{n}}ski$-Voigt numbers. We obtain fundamental combinatorial properties such as recurrence relations, generating functions and explicit formulas which are shown to be q-deformations of similar properties for the $Ruci{\acute{n}}ski$-Voigt numbers, and are generalizations of the results obtained by other authors. A combinatorial interpretation in the context of A-tableaux is also given where convolution-type identities are consequently obtained. Lastly, we establish the matrix decompositions of the $Ruci{\acute{n}}ski$-Voigt and the q-$Ruci{\acute{n}}ski$-Voigt numbers.
Keywords
Stirling number; $Ruci{\acute{n}}ski$-Voigt number; Whitney number; q-analogue; A-tabluea; matrix decomposition;
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1 R. B. Corcino and J. C. Fernandez, A combinatorial approach for q-analogue of r-Stirling numbers, British J. Math. Comput. Sci. 4 (2014), 1268-1279.   DOI
2 R. B. Corcino and M. M. Mangontarum, On multiparameter q-noncentral Stirling and Bell numbers, Ars Combin. 118 (2015), 201-220.
3 R. B. Corcino and C. B. Montero, A q-analogue of Rucinski-Voigt numbers, ISRN Discrete Math. 2012 (2012), Article ID 592818, 18 pages.
4 R. Ehrenborg, Determinants involving q-Stirling numbers, Adv. in Appl. Math. 31 (2003), no. 4, 630-642.   DOI
5 B. S. El-Desouky, R. S. Gomaa, and N. P. Cakic, q-analogues of multiparameter non- central Stirling and generalized harmonic numbers, Appl. Math. Comput. 232 (2014), 132-143.
6 H. W. Gould, The q-Stirling numbers of first and second kinds, Duke Math. J. 28 (1961), 281-289.   DOI
7 L. C. Hsu and P. J.-S. Shiue, A unified approach to generalized Stirling numbers, Adv. in Appl. Math. 20 (1998), no. 3, 366-384.   DOI
8 J. Katriel, Stirling number identities: interconsistency of q-analogues, J. Phys. A 31 (1998), no. 15, 3559-3572.   DOI
9 M.-S. Kim and J.-W. Son, A note on q-difference operators, Commun. Korean Math. Soc. 17 (2002), no. 3, 423-430.   DOI
10 R. B. Corcino, The (r, ${\beta}$)-Stirling numbers, The Mindanao Forum 14 (1999), 91-99.
11 M. Koutras, Noncentral Stirling numbers and some applications, Discrete Math. 42 (1982), no. 1, 73-89.   DOI
12 M. M. Mangontarum, Some theorems and applications of the (q, r)-Whitney numbers, J. Integer Seq. 20 (2017), no. 2, Art. 17.2.5, 26 pp.
13 M. M. Mangontarum, O. I. Cauntongan, and A. M. Dibagulun, A note on the translated Whitney numbers and their q-analogues, Turkish J. Anal. Number Theory 4 (2016), 74-81.
14 M. M. Mangontarum, O. I. Cauntongan, and A. P. Macodi-Ringia, The noncentral version of the Whitney numbers: a comprehensive study, Int. J. Math. Math. Sci. 2016 (2016), Art. ID 6206207, 16 pp.
15 M. M. Mangontarum and A. M. Dibagulun, On the translated Whitney numbers and their combinatorial properties, British J. Appl. Sci. Technology 11 (2015), 1-15.
16 M. M. Mangontarum and J. Katriel, On q-boson operators and q-analogues of the r- Whitney and r-Dowling numbers, J. Integer Seq. 18 (2015), no. 9, Article 15.9.8, 23 pp.
17 M. M. Mangontarum, A. P. Macodi-Ringia, and N. S. Abdulcarim, The translated Dowl- ing polynomials and numbers, International Scholarly Research Notices 2014 (2014), Article ID 678408, 8 pages.
18 A. de Medicis and P. Leroux, Generalized Stirling numbers, convolution formulae and p, q-analogues, Canad. J. Math. 47 (1995), no. 3, 474-499.   DOI
19 I. Mezo, A new formula for the Bernoulli polynomials, Results Math. 58 (2010), no. 3-4, 329-335.   DOI
20 J. Pan, Matrix decomposition of the unified generalized Stirling numbers and inversion of the generalized factorial matrices, J. Integer Seq. 15 (2012), no. 6, Article 12.6.6, 9 pp.
21 A. Rucinski and B. Voigt, A local limit theorem for generalized Stirling numbers, Rev. Roumaine Math. Pures Appl. 35 (1990), no. 2, 161-172.
22 L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948), 987-1000.   DOI
23 J. Stirling, Methodus Differentialissme Tractus de Summatione et Interpolatione Serierum Infinitarum, London, 1730.
24 A. Xu and T. Zhou, Some identities related to the r-Whitney numbers, Integral Transforms Spec. Funct. 27 (2016), no. 11, 920-929.   DOI
25 M. Arik and D. D. Coon, Hilbert spaces of analytic functions and generalized coherent states, J. Mathematical Phys. 17 (1976), no. 4, 524-527.   DOI
26 H. Belbachir and I. E. Bousbaa, Translated Whitney and r-Whitney numbers: a combinatorial approach, J. Integer Seq. 16 (2013), no. 8, Article 13.8.6, 7 pp.
27 M. Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.   DOI
28 M. Benoumhani, On some numbers related to Whitney numbers of Dowling lattices, Adv. in Appl. Math. 19 (1997), no. 1, 106-116.   DOI
29 A. Z. Broder, The r-Stirling numbers, Discrete Math. 49 (1984), no. 3, 241-259.   DOI
30 L. Carlitz, Weighted Stirling numbers of the first and second kind. I, Fibonacci Quart. 18 (1980), no. 2, 147-162.
31 J. Cigler, A new q-analog of Stirling numbers, Osterreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 201 (1992), no. 1-10, 97-109.
32 L. Comtet, Advanced Combinatorics, revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974.
33 R. B. Corcino, R. B. Corcino, J. M. Ontolan, C. M. Perez-Fernandez, and E. R. Cantallopez, The Hankel transform of q-noncentral Bell numbers, Int. J. Math. Math. Sci. 2015 (2015), Art. ID 417327, 10 pp.