On a q-Analogue of the Non-central Whitney Numbers

Omar I. Cauntongan

Abstract


In this paper, a q-analogue of the noncentral Whitney numbers of both kinds are define in terms of horizontal generating functions. Some properties such as recurrence relations, explicit formula, generating functions, orthogonality and inverse relations are established. Matrix decomposition of these q-analogues is presented in an explicit and non-recursive form. Moreover, a q-analogue of the noncentral Dowling numbers and polynomials are defining and establish some of their properties.


Keywords


Whitney numbers, noncentral Whitney numbers, Dowling numbers, non- central Dowling numbers, q-analogue

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References


Belbachir, H. and Bousbaa, I. E. (2013). Translated Whitney and r-Whitney Numbers: A Combinatorial Approach, J. Integer Sequences 16 (2013) Article 13.8.6.

Benoumhani, M. (1996) On Whitney numbers of Dowling Lattices, Discrete Math. 159 (1996),13-33.

Benoumhani, M.(1997). On Some Numbers Related to the Whitney Numbers of Dowling Lattices, Advances Appl. Math., 19 (1997), 106-116.

Broder, A.(1984). The r-Stirling numbers, Discrete Math. 49 (1984), 241-259.

Carlitz, L.(1984). q-Bernoulli numbers and Polynomials, Duke Math. J. Vol. 15 (1984),pp.987-1000.

Cheon, G.-S and Jung, J.-H. (2012). The r-Whitney Numbers of Dowling Lattices, Discrete Math., 312 (15) (2012), 2337-2348.

Conrad, K., A q-Analogue of Mahler Expansions I, Adv of Math Vol. 14(2), 1999.

Comtet, L.( 1974). Advanced Combinatorics, D. Reidel Publishing Company, 1974.

Corcino, R.B. (1999). The (r; β)-Stirling Numbers. Mindanao Forum. Vol. 14(2)

Corcino, R.B. and Barrientos, C. (2011). Some Theorems on the q-Analogue of the Generalized Stirling Numbers. Bulletin of the Malaysian Mathematical Sciences Society 34(3), (2011), 487-501.

Corcino, R.B., Hsu, L.C. and Tan, E.L. (2006). A q-Analogue of Generalized Stirling Numbers. The Fibonacci Quarterly, 44, (2006), 154-167.

Corcino, R.B. and Montero, C.B. (2006). A q-Analogue of Rucinski-Voigt numbers. ISRN Disc. Math., ARTICLE ID (592818) (2012) 18 pages.

Dowling, T. A. (1973). A Class of Geometric Lattices Based on Finite Groups, J. Combin.Theory, Ser. B, 15, (1973), 61-86.

Hsu, L. and Shiue, P. J. (1998). A Uni_ed Approach to Generalized Stirling Numbers, Advances in Applied Mathematics, 20 (1998), 366-384.

Gould, H.W. (1994). The q-Stirling Numbers of the First and Second Kinds. Duke Math. J. 28, (1994), 281-289.

Koutras, M. (1982). Non-central Stirling Numbers and Some Applications, Discrete Math. 42 (1982), 73-89.

Mangontarum, M., Cauntongan, O. and Macodi-Ringia A. (2016). The Noncentral Version of the Whitney Numbers: A Comprehensive Study. International Journal of Mathematics and Mathematical Sciences

Mangontarum, M., Macodi-Ringia A. and Abdulcarim, N., The Translated DowlingPolynomials and Numbers. International Scholarly Research Notices (2014) 8 pages.

Pan, J.( 2012). Matrix Decomposition of the Uni_ed Generalized Stirling Numbers and Inversion of the Generalized Factorial Matrices. Journal of integer sequences. Vol. 15(2012). pp.1-9.

Rahmani, M. (2014). Some Results on Whitney Numbers of Dowling Lattices, Arab Journal of Mathematics Sciences Vol. 20 No.2 (2014) pp. 11-27.

Stirling, J. (1749). Methodus Di_erentialissme Tractus de Summatione et Interpolatione Serierum In_nitarum, London, 1730. [English translation by F. Holliday with the titleThe Di_erential Method", London, 1749].


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