# The $r$-Bell numbers and matrices containing non-central Stirling and Lah numbers

Volume 19, Issue 3, pp 181--191 Publication Date: May 24, 2019       Article History
• 37 Downloads
• 92 Views

### Authors

Roberto B. Corcino - Research Institute for Computational Mathematics and Physics, Cebu Normal University, Osmena Boulevard, Cebu City, Philippines. Cristina B. Corcino - Research Institute for Computational Mathematics and Physics, Cebu Normal University, Osmena Boulevard, Cebu City, Philippines. Jeneveb T. Malusay - Research Institute for Computational Mathematics and Physics, Cebu Normal University, Osmena Boulevard, Cebu City, Philippines. Gaea Iolanthe Mari R. Bercero - Research Institute for Computational Mathematics and Physics, Cebu Normal University, Osmena Boulevard, Cebu City, Philippines.

### Abstract

In this paper, two new explicit formulas for $r$-Bell numbers are established. One formula is expressed in terms of $r$-Stirling numbers of the second kind and $r$-Lah numbers. The other formula is expressed in terms of the non-central Stirling numbers of the second kind and the ordinary Lah numbers. Moreover, some matrix relations are obtained involving $r$-Bell numbers, $r$-Stirling numbers of the second kind, $r$-Lah numbers, non-central Stirling numbers of the second kind, and the ordinary Lah numbers.

### Keywords

• $r$-Bell numbers
• $r$-Stirling numbers
• non-central Stirling numbers
• $r$-Lah numbers
• Lah numbers

•  05A15
•  11B65
•  11B73

### References

• [1] K. N. Boyadzhiev, Lah Numbers, Laguerre Polynomials of Order Negative One, and the $n$th Derivative of $\exp(1/x)$, Acta Univ. Sapientiae Math., 8 (2016), 22--31

• [2] A. Z. Broder, The $r$-Stirling numbers, Discrete Math., 49 (1984), 241--259

• [3] L. Comtet, Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht (1974)

• [4] R. B. Corcino, J. T. Malusay, J. Cillar, G. Rama, O. Silang, I. Tacoloy, Analogies of the Qi formula for some Dowling type numbers, arXiv (Accepted for Publication in Utilitas Mathematica), 2018 (2018), 22 pages

• [5] S. Daboul, J. Mangaldan, M. Z. Spivey, P. J. Taylor, The Lah numbers and the $n$--th derivative of $e^{{1}/{x}}$, Math. Mag., 86 (2013), 39--47

• [6] M. Koutras, Non-Central Stirling Numbers and Some Applications, Discrete Math., 42 (1982), 73--89

• [7] I. Mező, The $r$--Bell numbers, J. Integer Seq., 14 (2011), 14 pages

• [8] G. Nyul, G. Rácz, The $r$--Lah numbers, Discrete Math., 338 (2015), 1660--1666

• [9] F. Qi, An explicit formula for the Bell numbers in terms of Lah and Stirling numbers, Mediterr. J. Math., 13 (2016), 2795--2800