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

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## 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

- Belbachir, H., & Bousbaa, I. (2013). Translated Whitney and
*r*-Whitney numbers: A combinatorial approach, J. Integer Seq., 16, Article 13.8.6. - Chen, C., & Kho, K. (1992). Principles and Techniques in Combinatorics, World Scientific Publishing Co.
- Cillar, J. D., & Corcino, R. B. (2020). A
*q*-analogue of Qi formula for*r*-Dowling numbers, Commun. Korean Math. Soc., 35, 21–41. - Comtet, L. (1974). Advanced Combinatorics, D. Reidel Publishing Co.
- 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.
- 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. - 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.
- Garsia, A. M., & Remmel, J. (1980). A combinatorial interpretation of
*q*-derangement and*q*-Laguerre numbers, Europ. J. Combinatorics, 1, 47–59. - Graham, R. L., Knuth, D. E., & Patashnik, O. (1994). Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley.
- Guo, B., & Qi, F. (2015). Six proofs for an identity of the Lah numbers, Online J. Anal. Comb., 10, 5 pages.
- Kac, V., & Cheung, P. (2002). Quantum Calculus, Springer, New York, NY, USA.
- Lindsay, J., Mansour, T., & Shattuck, M. (2011). A new combinatorial interpretation of a
*q*-analogue of the Lah numbers, J. Comb., 2, 245–264. - Mangontarum, M. M., & Dibagulun, A. M. (2015). On the translated Whitney numbers and their combinatorial properties, British J. Appl. Sci. Technology, 11, 1–15.
- 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. - 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.
- Mansour, T., Mulay, S., & Shattuck, M. (2012). A general two-term recurrence and its solution, European J. Combin., 33, 20–26.
- Mansour, T., Ramırez, J. L., & Shattuck, M. (2017). A generalization of the
*r*-Whitney numbers of the second kind, J. Comb., 8, 29–55. - 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. - Mansour, T., & Shattuck, M. (2018). A generalized class of restricted Stirling and Lah numbers, Math. Slovaca, 68, 727–740.
- Petkovsek, M., & Pisanski, T. (2007). Combinatorial interpretation of unsigned Stirling and Lah numbers, Pi Mu Epsilon J., 12(7), 417–424.
- Qi, F. (2016). An explicit formula for the Bell numbers in terms of Lah and Stirling numbers, Mediterr. J. Math., 13, 2795–2800.

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## 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 .