A $q$-analogue of $\bar{\alpha }$-Whitney Numbers

We define the $(q,\bar{\mathit{\alpha }})$-Whitney numbers which are reduced to the $\bar{\mathit{\alpha }}$-Whitney numbers when $q\to 1$. Moreover, we obtain several properties of these numbers such as explicit formulas, recurrence relations, generating functions, orthogonality and inverse relations. Finally, we define the $\bar{\mathit{\alpha }}$-Whitney-Lah numbers as a generalization of the $r$-Whitney-Lah numbers and we introduce their important basic properties.