The generalized r-Whitney numbers

TL;DR

This paper introduces the generalized r-Whitney numbers, explores their properties, relations to Stirling numbers, and provides recurrence relations, generating functions, and matrix representations for these new combinatorial numbers.

Contribution

The paper defines and analyzes the generalized r-Whitney numbers, establishing their relations to Stirling numbers and deriving their recurrence relations and generating functions.

Findings

Derived recurrence relations and generating functions for the generalized r-Whitney numbers.

Established relations between generalized Whitney and Stirling numbers.

Provided matrix representations of these relationships.

Abstract

In this paper, we define the generalized r-Whitney numbers of the first and second kind. Moreover, we drive the generalized Whitney numbers of the first and second kind. The recurrence relations and the generating functions of these numbers are derived. The relations between these numbers and generalized Stirling numbers of the first and second kind are deduced. Furthermore, some special cases are given. Finally, matrix representation of The relations between Whitney and Stirling numbers are given.