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On ***q*-Boson Operators and *q*-Analogues of the *r*-Whitney and *r*-Dowling Numbers

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Mahid M. Mangontarum

Department of Mathematics

Mindanao State University – Main Campus

Marawi City 9700

Philippines

Jacob Katriel

Department of Chemistry

Technion – Israel Institute of Technology

Haifa 32000

Israel

**Abstract:**

We define the (*q*, *r*)-Whitney numbers of the first and second kinds in
terms of the *q*-Boson operators, and obtain several fundamental
properties such as recurrence formulas, orthogonality and inverse
relations, and other interesting identities. As a special case, we
obtain a *q*-analogue of the *r*-Stirling numbers of the first and second
kinds. Finally, we define the (*q*, *r*)-Dowling polynomials in terms of
sums of (*q*, *r*)-Whitney numbers of the second kind, and obtain some of
their properties.

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(Concerned with sequences
A000110
A003575
A008275
A008277.)

Received May 13 2015; revised version received, September 1 2015.
Published in *Journal of Integer Sequences*, September 7 2015.

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