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In this paper, we establish an explicit formula for $r$-Dowling numbers in terms of $r$-Whitney Lah and $r$-Whitney numbers of the second kind. This is a generalization of the Qi formula for Bell numbers in terms of Lah and Stirling numbers of the second kind. Moreover, we define the $q,r$-Dowling numbers, $q,r$-Whitney Lah numbers and $q,r$-Whitney numbers of the first kind and obtain several fundamental properties of these numbers such as orthogonality and inverse relations, recurrence relations, and generating functions. Hence, we derive an analogous Qi formula for $q,r$-Dowling numbers expressed in terms of $q,r$-Whitney Lah numbers and $q,r$-Whitney numbers of the second kind.
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참고문헌 (12)
참고문헌 신청
M. Benoumhani / 1996 / On Whitney numbers of Dowling lattices / Discrete Math 159(1-3):13~33
L. Carlitz / 1948 / q-Bernoulli numbers and polynomials / Duke Math. J. 15:987~1000
G. -S. Cheon / 2012 / r-Whitney numbers of Dowling lattices / Discrete Math 312(15):2337~2348
L. Comtet / 1974 / Advanced Combinatorics / D. Reidel Publishing Co
R. Corcino / 2019 / Analogies of the Qi formula for some Dowling type numbers / Util. Math 111:3~26
L. Carlitz / 1948 / q-Bernoulli numbers and polynomials / Duke Math. J. 15:987~1000
G. -S. Cheon / 2012 / r-Whitney numbers of Dowling lattices / Discrete Math 312(15):2337~2348
L. Comtet / 1974 / Advanced Combinatorics / D. Reidel Publishing Co
R. Corcino / 2019 / Analogies of the Qi formula for some Dowling type numbers / Util. Math 111:3~26
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