Bell Numbers, Log-Concavity, and Log-Convexity

Asai, Nobuhiro; Kubo, Izumi; Kuo, Hui-Hsiung

doi:10.1007/978-94-010-0842-6_4

Cited by 18 publications

(27 citation statements)

References 6 publications

(11 reference statements)

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“…On the other hand, log-convexity has so far attracted much less attention. Results appear scattered in the literature, mixed with other stuff [1,5], or as footnotes in log-concavity oriented papers. In particular, there is no paper that presents and compares various methods of establishing log-convexity.…”

Section: Logarithmic Behavior Of Sequencesmentioning

confidence: 88%

“…For example, on segment [3,4] the function f is given by f (x) = 7x−1 2(x+2) , and on the segment [4,5] it is given by f (x) = 20x 2 −9x−14 (x+2)(7x −8) . Obviously, the values of function f at positive integers coincide with the elements of the quotient sequence, f (n) = x n .…”

Section: Calculus Method-motzkin Numbersmentioning

confidence: 99%

“…A nice example is the recent use of log-convex sequences in quantum physics for constructing generalized coherent states in models with discrete nonlinear spectra [25,36]. Then, the log-convexity of certain sequences (of generalized Bell numbers) enabled their use as important examples in white noise theory [5,6,27]. The log-convex sequences also play an important role in probability-the log-convexity of a sequence P {X = n} is a sufficient condition for a discrete random variable X to be infinitelydivisible [30,53].…”

Section: Introductionmentioning

confidence: 98%

See 2 more Smart Citations

Seven (Lattice) Paths to Log-Convexity

Došlić

2009

Acta Appl Math

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Section: Logarithmic Behavior Of Sequencesmentioning

confidence: 88%

Section: Calculus Method-motzkin Numbersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Seven (Lattice) Paths to Log-Convexity

Došlić

2009

Acta Appl Math

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“…This consideration is quite important to discuss the continuity of various operators, Wick products and so on. See also [2][18] [23]. This matter is not addressed in the paper by GHOR.…”

Section: Characterization Theorems and Quantum White Noisesmentioning

confidence: 99%

“…We remark that we need the condition in (H4) in order to discuss quantum Poisson white noise and exponential vector associated with Poisson measure given by (3.2). In addition, (H4) will be necessary to prove Theorem 4.5 (2).…”

Section: Technical Details Of Equationsmentioning

confidence: 99%

Gaussian and Poisson white noises with related characterization theorems

Asai¹,

Kubo²,

Kuo³

2003

Finite and Infinite Dimensional Analysis in Honor of Leonard Gross

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Dedicated to Professor Leonard Gross on the occasion of his 70th birthday.Abstract. Let µ G and µ P be a Gaussian measure and a Poisson measure on E * , respectively. Let at and a * t be respectively annihilation and creation operators at a point t ∈ R. In the theory of quantum white noise, it is known that at is a continuous linear operator from Γu(E C ) into itself and a * t is a continuous linear operator from Γu(E C ) * into itself. In paticular, at + a * t and at+a * t +a * t at+I are called the quantum Gaussian white noise and the quantum Poisson white noise, respectively. The main purpose of this work is to realize quantum Gaussian and Poisson white noises in terms of multiple Wiener-Itô integrals, and show that such realizations cannot be achieved by J-transform and its holomorphy, but can be done by S X -transform depending on the exponential function φ X ξ , which determines a unitary isomorphism between Boson Fock space and L 2 (E * , µ X ), X = G, P . In Appendix A, some connections between [6][7] and [9] will be discussed.

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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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Bell Numbers, Log-Concavity, and Log-Convexity

Abstract: n=0 is log-concave and the sequence {b k (n)} ∞ n=0 is log-convex, or equivalently, the following inequalities hold for all n ≥ 0,

Cited by 18 publications

References 6 publications

Seven (Lattice) Paths to Log-Convexity

Seven (Lattice) Paths to Log-Convexity

Gaussian and Poisson white noises with related characterization theorems

Some properties and an application of multivariate exponential polynomials

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