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

TL;DR

This paper introduces explicit formulas and fundamental properties for (q,r)-Whitney-Lah numbers and (q,r)-Dowling numbers, extending classical combinatorial numbers through q-analogues and interpolation techniques.

Contribution

It derives explicit formulas for (q,r)-Whitney-Lah numbers and (q,r)-Dowling numbers using q-difference operators and Newton's interpolation, advancing the understanding of these q-analogues.

Findings

Explicit formulas for (q,r)-Whitney-Lah numbers derived.

Fundamental recurrence relations established.

Explicit formula for (q,r)-Dowling numbers obtained.

Abstract

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 q-analogue are established such as vertical and horizontal recurrence relations, horizontal and exponential generating functions. Moreover, an explicit formula for (q,r)-Whitney-Lah number is derived using the concept of q-difference operator, particularly, the q-analogue of Newton's Interpolation Formula (the umbral version of Taylor series). Furthermore, an explicit formula for the first form (q,r)-Dowling numbers is obtained which is expressed in terms of (q,r)-Whitney-Lah numbers and (q,r)-Whitney numbers of the second kind.