Bivariate Extension of the r-Dowling Polynomials and the Generalized Spivey's Formula

TL;DR

This paper extends r-Dowling polynomials to bivariate forms, generalizes properties of bivariate Bell and r-Bell polynomials, and derives two generalized Spivey's formulas.

Contribution

It introduces a bivariate extension of r-Dowling polynomials and generalizes classical combinatorial identities.

Findings

Established properties of the bivariate r-Dowling polynomials.

Derived two generalized forms of Spivey's formula.

Connected the new polynomials to bivariate Bell and r-Bell polynomials.

Abstract

In this paper, we extend the r-Dowling polynomials to their bivariate forms. Several properties that generalize those of the bivariate Bell and r-Bell polynomials are established. Finally, we obtain two forms of generalized Spivey's formula.