Some combinatorial identities for the r-Dowling polynomials

Mark Shattuck
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 2, Pages 145-154
DOI: 10.7546/nntdm.2019.25.2.145-154
Full paper (PDF, 201 Kb)

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Authors and affiliations

Mark Shattuck
Institute for Computational Science & Faculty of Mathematics and Statistics
Ton Duc Thang University, Ho Chi Minh City, Vietnam

Abstract

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 aⁱ 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.

Keywords

• Bell numbers
• r-Dowling polynomials
• r-Whitney numbers
• Polynomial generalization

2010 Mathematics Subject Classification

• 05A19
• 11B73

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