Journal of the
Korean Mathematical Society JKMS

In this paper, we define a $p,q$-difference operator and obtain an explicit formula which is used to express the $p,q$-analogue of the unified generalization of Stirling numbers and its exponential generating function in terms of the $p,q$-difference operator. Explicit formulas for the non-central $q$-Stirling numbers of the second kind and non-central $q$-Lah numbers are derived using the new $q$-analogue of Newton's interpolation formula. Moreover, a $p,q$-analogue of Newton's interpolation formula is established.

Keywords: $p,q$-difference operator, $q$-Stirling numbers, $q$-Lah numbers, Newton's interpolation formula, exponential generating function

MSC numbers: 05A30, 05A15, 11B65, 11B73