A $q$-Analogue of $r$-Whitney Numbers of the Second Kind and Its Hankel Transform

A $q$-analogue of $r$-Whitney numbers of the second kind, denoted by $W_{m,r}[n,k]_q$, is defined by means of a triangular recurrence relation. In this paper, several fundamental properties for the $q$-analogue are established including other forms of recurrence relations, explicit formulas and generating functions. Moreover, a kind of Hankel transform for $W_{m,r}[n,k]_q$ is obtained.