  In this article we generalize the $q$-difference operator due to Carlitz in
order to derive explicit sum formulae for several extensions of Stirling
numbers of the second kind, including complete homogeneous symmetric functions,
complementary symmetric functions, $r$-Whitney numbers and elliptic analogues
of rook, Stirling and Lah numbers. Furthermore, we generalize Carlitz'
$q$-Eulerian numbers to a Lagrange polynomial extension. We define them by
generalizing Worpitzky's identity appropriately, and derive a recursion and an
explicit sum formulae. Special cases include $r$-Whitney Eulerian numbers and
elliptic Eulerian numbers.
