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.

In this paper, we introduce a class of truncated tangent polynomials which generalizes tangent numbers and polynomials, and establish various properties and identities. Moreover, we obtain some interesting correlations of truncated tangent polynomials with the Stirling numbers of the second kind and with the hypergeometric Bernoulli polynomials.