The translated Whitney–Lah numbers: generalizations and q-analogues

Mahid M. Mangontarum
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 4, Pages 80—92
DOI: 10.7546/nntdm.2020.26.4.80-92
Download full paper: PDF, 203 Kb

Details

Authors and affiliations

Mahid M. Mangontarum
Department of Mathematics, Mindanao State University–Main Campus
Marawi City 9700, Philippines

Abstract

In this paper, we derive some combinatorial formulas for the translated Whitney–Lah numbers which are found to be generalizations of already-existing identities of the classical Lah numbers, including the well-known Qi’s formula. Moreover, we obtain q-analogues of the said formulas and identities by establishing similar properties for the translated q-Whitney numbers.

Keywords

  • Lah numbers
  • translated Whitney–Lah numbers
  • Qi’s formula
  • q-analogues.

2010 Mathematics Subject Classification

  • 05A19
  • 05A30
  • 11B65

References

  1.  Belbachir, H., & Bousbaa, I. (2013). Translated Whitney and r-Whitney numbers: A combinatorial approach, J. Integer Seq., 16, Article 13.8.6.
  2. Chen, C., & Kho, K. (1992). Principles and Techniques in Combinatorics, World Scientific Publishing Co.
  3. Cillar, J. D., & Corcino, R. B. (2020). A q-analogue of Qi formula for r-Dowling numbers, Commun. Korean Math. Soc., 35, 21–41.
  4. Comtet, L. (1974). Advanced Combinatorics, D. Reidel Publishing Co.
  5. Corcino, R. B., Malusay, J. T., Cillar, J. D., Rama, G., Silang, O., & Tacoloy, I. (2019). Analogies of the Qi formula for some Dowling type numbers, Util. Math., 111, 3–26.
  6. Corcino, R. B., Montero, C. B, Montero, M. B., & Ontolan, J. M. (2019). The r-Dowling numbers and matrices containing r-Whitney numbers of the second kind and Lah numbers, Eur. J. Pure Appl. Math., 12, 1122–1137.
  7.  Daboul, S., Mangaldan, J., Spivey, M. Z., & Taylor, P. J. (2013). The Lah numbers and the nth derivative of e1=x, Math. Magazine, 86, 39–47.
  8. Garsia, A. M., & Remmel, J. (1980). A combinatorial interpretation of q-derangement and q-Laguerre numbers, Europ. J. Combinatorics, 1, 47–59.
  9. Graham, R. L., Knuth, D. E., & Patashnik, O. (1994). Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley.
  10. Guo, B., & Qi, F. (2015). Six proofs for an identity of the Lah numbers, Online J. Anal. Comb., 10, 5 pages.
  11. Kac, V., & Cheung, P. (2002). Quantum Calculus, Springer, New York, NY, USA.
  12. Lindsay, J., Mansour, T., & Shattuck, M. (2011). A new combinatorial interpretation of a q-analogue of the Lah numbers, J. Comb., 2, 245–264.
  13. Mangontarum, M. M., & Dibagulun, A. M. (2015). On the translated Whitney numbers and their combinatorial properties, British J. Appl. Sci. Technology, 11, 1–15.
  14. Mangontarum, M. M., Cauntongan, O. I., & Dibagulun, A. M. (2016). A note on the translated Whitney numbers and their q-analogues, Turkish Journal of Analysis and Number Theory, 4, 74–81.
  15. Mangontarum, M. M., Macodi-Ringia, A. P., & Abdulcarim, N. S. (2014). The translated Dowling polynomials and numbers, International Scholarly Research Notices, 2014, Article ID 678408, 8 pages.
  16. Mansour, T., Mulay, S., & Shattuck, M. (2012). A general two-term recurrence and its solution, European J. Combin., 33, 20–26.
  17. Mansour, T., Ramırez, J. L., & Shattuck, M. (2017). A generalization of the r-Whitney numbers of the second kind, J. Comb., 8, 29–55.
  18. Mansour, T., Ramırez, J. L., Shattuck, M., & Villamarin, S. N. (2019). Some combinatorial identities of the r-Whitney-Eulerian numbers, Appl. Anal. Discrete Math., 13, 378–398.
  19. Mansour, T., & Shattuck, M. (2018). A generalized class of restricted Stirling and Lah numbers, Math. Slovaca, 68, 727–740.
  20. Petkovsek, M., & Pisanski, T. (2007). Combinatorial interpretation of unsigned Stirling and Lah numbers, Pi Mu Epsilon J., 12(7), 417–424.
  21. Qi, F. (2016). An explicit formula for the Bell numbers in terms of Lah and Stirling numbers, Mediterr. J. Math., 13, 2795–2800.

Related papers

Cite this paper

Mangontarum, M. M. (2020). The translated Whitney–Lah numbers: generalizations and q-analogues. Notes on Number Theory and Discrete Mathematics, 26 (4), 80-92, doi: 10.7546/nntdm.2020.26.4.80-92  .

Comments are closed.