In this paper, we prove by algebraic methods two further formulas satisfied by the r-Whitney numbers of the second kind, which have been an object of recent study. We establish a version of one of the formulas for a polynomial generalization of the r-Whitney numbers. As special cases, we obtain apparently new identities for the Stirling numbers of the second kind and the Bell numbers. Moreover, one may obtain analogous formulas for Lah numbers and Stirling numbers of the first kind.