The r-Dowling Numbers and Matrices Containing r-Whitney Numbers of the Second Kind and Lah Numbers
2019 ◽
Vol 12
(3)
◽
pp. 1122-1137
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.
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Vol 14
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pp. 65-81
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pp. 367-379
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