Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A q-ANALOGUE OF QI FORMULA FOR r-DOWLING NUMBERS

  • Cillar, Joy Antonette D. (Department of Mathematics and Sciences University of San Jose-Recoletos) ;
  • Corcino, Roberto B. (Research Institute for Computational Mathematics and Physics Cebu Normal University)
  • Received : 2018.11.19
  • Accepted : 2019.03.13
  • Published : 2020.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we establish an explicit formula for r-Dowling numbers in terms of r-Whitney Lah and r-Whitney numbers of the second kind. This is a generalization of the Qi formula for Bell numbers in terms of Lah and Stirling numbers of the second kind. Moreover, we define the q, r-Dowling numbers, q, r-Whitney Lah numbers and q, r-Whitney numbers of the first kind and obtain several fundamental properties of these numbers such as orthogonality and inverse relations, recurrence relations, and generating functions. Hence, we derive an analogous Qi formula for q, r-Dowling numbers expressed in terms of q, r-Whitney Lah numbers and q, r-Whitney numbers of the second kind.

Keywords

References

  1. M. Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33. https://doi.org/10.1016/0012-365X(95)00095-E
  2. L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948), 987-1000. http://projecteuclid.org/euclid.dmj/1077475200 https://doi.org/10.1215/S0012-7094-48-01588-9
  3. G.-S. Cheon and J.-H. Jung, r-Whitney numbers of Dowling lattices, Discrete Math. 312 (2012), no. 15, 2337-2348. https://doi.org/10.1016/j.disc.2012.04.001
  4. L. Comtet, Advanced Combinatorics, revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974.
  5. R. Corcino, J. T. Malusay, J. Cillar, G. Rama, O. Silang, and I. Tacoloy, Analogies of the Qi formula for some Dowling type numbers, Util. Math. 111 (2019), 3-26.
  6. R. Corcino and C. Montero, A q-analogue of Rucinski-Voigt numbers, ISRN Discrete Math. 2012 (2012), Article ID 592818, 18 pages. https://doi.org/10.5402/2012/592818
  7. E. Gyimesi and G. Nyul, A comprehensive study of r-Dowling polynomials, Aequationes Math. 92 (2018), no. 3, 515-527. https://doi.org/10.1007/s00010-017-0538-z
  8. E. Gyimesi and G. Nyul, New combinatorial interpretations of r-Whitney and r-Whitney-Lah numbers, Discrete Appl. Math. 255 (2019), 222-233. https://doi.org/10.1016/j.dam.2018.08.020
  9. I. Mezo, A new formula for the Bernoulli polynomials, Results Math. 58 (2010), no. 3-4, 329-335. https://doi.org/10.1007/s00025-010-0039-z
  10. F. Qi, An explicit formula for the Bell numbers in terms of the Lah and Stirling numbers, Mediterr. J. Math. 13 (2016), no. 5, 2795-2800. https://doi.org/10.1007/s00009-015-0655-7
  11. J. L. Ramirez and M. Shattuck, A (p, q)-analogue of the r-Whitney-Lah numbers, J. Integer Seq. 19 (2016), no. 5, Article 16.5.6, 21 pp.
  12. A. Rucinski and B. Voigt, A local limit theorem for generalized Stirling numbers, Rev. Roumaine Math. Pures Appl. 35 (1990), no. 2, 161-172.