The \(r\)-Bell numbers and matrices containing non-central Stirling and Lah numbers

Volume 19, Issue 3, pp 181--191 http://dx.doi.org/10.22436/jmcs.019.03.05
Publication Date: May 24, 2019 Submission Date: April 15, 2019 Revision Date: April 25, 2019 Accteptance Date: May 02, 2019

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.


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