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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## Details

### 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 *a ^{i}* and

*b*may be replaced by arbitrary sequences of variables

^{j}*x*and

_{i}*y*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.

_{j}### Keywords

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

### 2010 Mathematics Subject Classification

- 05A19
- 11B73

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## Cite this paper

APAShattuck, 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.

Shattuck, M. “Some combinatorial identities for the *r*-Dowling polynomials.” Notes on Number Theory and Discrete Mathematics 25, no. 2 (2019): 145-154, doi: 10.7546/nntdm.2019.25.2.145-154.

Shattuck, M. “Some combinatorial identities for the *r*-Dowling polynomials.” Notes on Number Theory and Discrete Mathematics 25.2 (2019): 145-154. Print, doi: 10.7546/nntdm.2019.25.2.145-154.