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
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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


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 bj may be replaced by arbitrary sequences of variables xi and yj 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.


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

2010 Mathematics Subject Classification

  • 05A19
  • 11B73


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Cite this paper

Shattuck, M.(2019). Some combinatorial identities for the r-Dowling polynomials. Notes on Number Theory and Discrete Mathematics, 25(2), 145-154, doi: 10.7546/nntdm.2019.25.2.145-154.

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