Generalized Bell polynomials

In this paper, generalized Bell polynomials (bnϕ)n associated to a sequence of real numbers ϕ=(ϕi)i=1∞ are introduced. Bell polynomials correspond to ϕi=0, i≥1. We prove that when ϕi≥0, i≥1: (a) the zeros of the generalized Bell polynomial bnϕ are simple, real and non positive; (b) the zeros of bn+1...

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Bibliographic Details
Published in:Journal of approximation theory 2025-03, Vol.306, p.106121, Article 106121
Main Author: Durán, Antonio J.
Format: Article
Language:English
Subjects:
Bell polynomials
Interlacing
Laguerre multiple polynomials
Zeros
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Summary:In this paper, generalized Bell polynomials (bnϕ)n associated to a sequence of real numbers ϕ=(ϕi)i=1∞ are introduced. Bell polynomials correspond to ϕi=0, i≥1. We prove that when ϕi≥0, i≥1: (a) the zeros of the generalized Bell polynomial bnϕ are simple, real and non positive; (b) the zeros of bn+1ϕ interlace the zeros of bnϕ; (c) the zeros are decreasing functions of the parameters ϕi. We find a hypergeometric representation for the generalized Bell polynomials. As a consequence, it is proved that the class of all generalized Bell polynomials is actually the same class as that of all Laguerre multiple polynomials of the first kind.
ISSN:0021-9045
DOI:10.1016/j.jat.2024.106121