Explicit Formulas for the First Form (q,r)-Dowling Numbers and (q,r)-Whitney-Lah Numbers

In this paper, a q-analogue of r-Whitney-Lah numbers, also known as (q,r)-Whitney-Lah number, denoted by $L_{m,r} [n, k]_q$ is defined using the triangular recurrence relation. Several fundamental properties for the are established such vertical and horizontal relations, exponential generating functions. Moreover, an explicit formula (q, r)-Whitney-Lah number derived concept q-difference operator, particularly, of Newton’s Interpolation Formula (the umbral version Taylor series). Furthermore, first form r)-Dowling numbers obtained which expressed in terms (q,r)-Whitney second kind.