# A $q$-analogue of $\bar\alpha$-Whitney Numbers

## TL;DR

This paper introduces a $q$-analogue of $ar\alpha$-Whitney numbers, exploring their properties and generalizations, including explicit formulas, recurrence relations, and orthogonality, extending classical combinatorial numbers.

## Contribution

The paper defines the $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers and their properties, and introduces the $ar{\boldsymbol{\alpha}}$-Whitney-Lah numbers as a new generalization.

## Key findings

- Derived explicit formulas and recurrence relations.
- Established orthogonality and inverse relations.
- Defined and analyzed the properties of Whitney-Lah numbers.

## Abstract

We define the $(q,\bar{\boldsymbol{\alpha}})$-Whitney numbers which are reduced to the $\bar{\boldsymbol{\alpha}}$-Whitney numbers when $q\rightarrow1$. Moreover, we obtain several properties of these numbers such as explicit formulas, recurrence relations, generating functions, orthogonality and inverse relations. Finally, we define the $\bar{\boldsymbol{\alpha}}$-Whitney-Lah numbers as a generalization of the $r$-Whitney-Lah numbers and we introduce their important basic properties.