Generalizations of Bell number formulas of spivey and Mező | Zendy

Mark Shattuck | Zendy

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Generalizations of Bell number formulas of spivey and Mező

Author(s) -

Mark Shattuck

Publication year - 2016

Publication title -

filomat

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.449

H-Index - 34

eISSN - 2406-0933

pISSN - 0354-5180

DOI - 10.2298/fil1610683s

Subject(s) - stirling numbers of the second kind , stirling number , mathematics , binomial coefficient , stirling numbers of the first kind , bell polynomials , extension (predicate logic) , binomial (polynomial) , binomial theorem , interpretation (philosophy) , generating function , discrete mathematics , binomial distribution , central binomial coefficient , combinatorics , negative binomial distribution , pure mathematics , statistics , poisson distribution , computer science , programming language

We provide q -generalizations of Spivey’s Bell number formula in various settings by considering statistics on different combinatorial structures. This leads to new identities involving q -Stirling numbers of both kinds and q -Lah numbers. As corollaries, we obtain identities for both binomial and q -binomial coefficients. Our results at the same time also generalize recent r -Stirling number formulas of Mezo. Finally, we provide a combinatorial proof and refinement of Xu’s extension of Spivey’s formula to the generalized Stirling numbers of Hsu and Shiue. To do so, we develop a combinatorial interpretation for these numbers in terms of extended Lah distributions.

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