A ( p,q )-analogue of Qi-type formula for r -Dowling numbers

Abstract

In this paper, \((p,q)\)-analogues of \(r\)-Whitney numbers of the first and second kinds are defined using horizontal generating functions. Several fundamental properties such as orthogonality and inverse relations, an explicit formula, and a kind of exponential generating function are obtained. Moreover, a \((p,q)\)-analogue of \(r\)-Whitney-Lah numbers is also defined in terms of a horizontal generating function, where necessary properties are obtained. These properties help develop a \((p,q)\)-analogue of the \(r\)-Dowling numbers, particularly, a \((p,q)\)-analogue of a Qi-type formula.