The r-Dowling Numbers and Matrices Containing r-Whitney Numbers of the Second Kind and Lah Numbers

Roberto Bagsarsa Corcino, Charles Montero, Maribeth Montero, Jay Ontolan

Abstract

This paper derives another form of explicit formula for $(r,\beta)$-Bell numbers using the Faa di Bruno's formula and certain identity of Bell polynomials of the second kind. This formula is expressed in terms  of the $r$-Whitney numbers of the second kind and the ordinary Lah numbers. As a consequence, a relation between $(r,\beta)$-Bell numbers and the sums of row entries of the product of two matrices containing the $r$-Whitney numbers of the second kind and the ordinary Lah numbers is established.  Moreover, a $q$-analogue of the explicit formula is obtained.

Keywords

$r$-Dowling numbers, $(r,\beta)$-Bell numbers, Bell polynomials, Lah numbers, $r$-Whitney numbers, Faa di Bruno's formula, $r$-Whitney-Lah numbers

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