Some combinatorial identities for the r-Dowling polynomials

Recently, three new Bell number formulas were proven using algebraic methods, one of which extended an earlier identity of Gould-Quaintance and another a previous identity of Spivey. Here, making use of combinatorial arguments to establish our results, we find generalizations of these formulas in terms of the r-Dowling polynomials. In two cases, weights of the form a(i) and b(j) may be replaced by arbitrary sequences of variables x(i) and y(j) which yields further generalizations. Finally, a second extension of one of the formulas is found that involves generalized Stirling polynomials and leads to analogues of this formula for other counting sequences.