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

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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