Higher order bell polynomials and the relevant integer sequences

Natalini, Pierpaolo; Ricci, Paolo Emilio

doi:10.2298/aadm1702327n

Cited by 13 publications

(11 citation statements)

References 10 publications

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“…One of anonymous referees pointed out that the formula (1.8) is essentially the same as those formulas in [26,Section 4]. The difference is that, in the article [26], the expressions are given, in compact form, in terms of the Hermite-Kampé de Fériet polynomials. By using definitions of these classical polynomials [3], the explicit forms given in (1.8) can be recovered.…”

Section: Power Functionmentioning

confidence: 94%

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Special values of the Bell polynomials of the second kind for some sequences and functions

Niu

Lim

et al. 2020

Journal of Mathematical Analysis and Applications

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In the paper, after concisely surveying some closed formulas and applications of special values of the Bell polynomials of the second kind for some special sequences and elementary functions, the authors newly establish some closed formulas for some special values of the Bell polynomials of the second kind. 4. Remarks and comparisons 21 Funding 24 Acknowledgements 24 References 24 In this paper, we have three aims which can be concretely stated as follows.

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Section: Power Functionmentioning

confidence: 94%

“…There have been a number of literature, such as [1,6,7,20,22,23,26,27,102,103,104,105], dedicated to the general theory of the Bell polynomials of the second kind B n,k (x 1 , x 2 , . .…”

Section: Survey Of Closed Formulas For the Bell Polynomialsmentioning

confidence: 99%

Special values of the Bell polynomials of the second kind for some sequences and functions

Niu

Lim

et al. 2020

Journal of Mathematical Analysis and Applications

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“…Remark In recent years, there have been some literature such as in previous studies devoted to deep investigation and extensive applications of the Bell polynomials of the second kind

B_{n, k} false(x_{n - k + 1} false)

.…”

Section: An Explicit Formula An Identity and Three Recurrence Relatmentioning

confidence: 99%

Some properties and an application of multivariate exponential polynomials

Niu

Lim

et al. 2020

Math Methods in App Sciences

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In the paper, the authors introduce a notion “multivariate exponential polynomials” which generalize exponential numbers and polynomials, establish explicit formulas, inversion formulas, and recurrence relations for multivariate exponential polynomials in terms of the Stirling numbers of the first and second kinds with the help of the Faà di Bruno formula, two identities for the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds, construct some determinantal inequalities and product inequalities for multivariate exponential polynomials with the aid of some properties of completely monotonic functions and other known results, derive the logarithmic convexity and logarithmic concavity for multivariate exponential polynomials, and finally find an application of multivariate exponential polynomials to white noise distribution theory by confirming that multivariate exponential polynomials satisfy conditions for sequences required in white noise distribution theory.

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“…The higher order Bell numbers, also known as higher order exponential numbers, have been considered in [5,7,22], and used in [2] in the framework of Brenke and Sheffer polynomials.…”

Section: Remarkmentioning

confidence: 99%

New Bell–Sheffer Polynomial Sets

Natalini

Ricci

2018

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In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them, have been studied. The method used in previous articles, and even in the present one, traces back to preceding results by Dattoli and Ben Cheikh on the monomiality principle, showing the possibility to derive explicitly the main properties of Sheffer polynomial families starting from the basic elements of their generating functions. The introduction of iterated exponential and logarithmic functions allows to construct new sets of Bell–Sheffer polynomials which exhibit an iterative character of the obtained shift operators and differential equations. In this context, it is possible, for every integer r, to define polynomials of higher type, which are linked to the higher order Bell-exponential and logarithmic numbers introduced in preceding papers. Connections with integer sequences appearing in Combinatorial analysis are also mentioned. Naturally, the considered technique can also be used in similar frameworks, where the iteration of exponential and logarithmic functions appear.

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Higher order bell polynomials and the relevant integer sequences

Cited by 13 publications

References 10 publications

Special values of the Bell polynomials of the second kind for some sequences and functions

Special values of the Bell polynomials of the second kind for some sequences and functions

Some properties and an application of multivariate exponential polynomials

New Bell–Sheffer Polynomial Sets

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