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The \((r_{1},\dots ,r_{p})\)-Bell polynomials. (English) Zbl 1308.11027

The paper is a continuation of research undertaken by the authors in their previous paper concerning the properties of the \((r_1, r_2, ..., r_p)\)-Stirling numbers of the second kind and the \(r := (r_1, r_2, ..., r_p)\)-Bell polynomials \(B_n(z;r)\). Let us recall that these numbers and polynomials – introduced by the authors in 2012 – represent the generalization of the respective numbers and polynomials defined earlier by A. Z. Broder [Discrete Math. 49, 241–259 (1984; Zbl 0535.05006)] and I. Mezö [Adv. Appl. Math. 41, No. 3, 293–306 (2008; Zbl 1165.11023); J. Integer Seq. 14, No. 1, Article 11.1.1, 14 p. (2011; Zbl 1205.05017)], respectively. With no doubts, the \((r_1, r_2, ..., r_p)\)-Stirling numbers of the second kind and the \((r_1, r_2, ..., r_p)\)-Bell polynomials are determined very well, from the combinatoric as well as from the analytic point of view, they were created as the natural generalization of the classic Stirling numbers of the second kind and the Bell polynomials.
In present paper the authors prove many new properties of \((r_1, r_2, ..., r_p)\)-Stirling numbers and the \((r_1, r_2, ..., r_p)\)-Bell polynomials. Among others, they prove that the roots of polynomials \(B_n(z;r)\) are real and non-positive which implies, by Newton’s inequality, the log-concavity (and thus the unimodality) of the respective finite sequence of \((r_1, r_2, ..., r_p)\)-Stirling numbers. Moreover, by applying the Darroch’s inequality (version of this inequality for polynomials, not included in the original cited Darroch’s paper, is considered here) the authors find the maximal element in this sequence. Furthermore, thy present many recurrence relations and generating functions related to the \((r_1, r_2, ..., r_p)\)-Bell polynomials.
Summing up, the paper is well edited, the authors take their art with passion. For sure it is worth to recommend this paper to the readers, and not only to the experts in the presented subject matter.

MSC:

11B73 Bell and Stirling numbers
05A10 Factorials, binomial coefficients, combinatorial functions
11B83 Special sequences and polynomials
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