Convergence properties of the q-deformed binomial distribution

Zeiner, Martin

doi:10.2298/aadm1000016z

Cited by 2 publications

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References 5 publications

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“…Since its introduction [15], this distribution has been extensively studied. For example, its generating function is given in [15], and its mean can be expressed in terms of the q-Pochhammer symbols and q-binomial coefficients [19]. Mathematically, it is referred to as the q-Bernstein basis function, in connection with q-Bernstein polynomials [17,18].…”

Section: Probability Distribution Of Total Change Of Spin Polarimentioning

confidence: 99%

Landau-Zener extension of the Tavis-Cummings model: Structure of the solution

Sun

Sinitsyn

2016

Phys. Rev. A

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We explore the recently discovered solution of the driven Tavis-Cummings model (DTCM). It describes interaction of arbitrary number of two-level systems with a bosonic mode that has linearly time-dependent frequency. We derive compact and tractable expressions for transition probabilities in terms of the well known special functions. In the new form, our formulas are suitable for fast numerical calculations and analytical approximations. As an application, we obtain the semiclassical limit of the exact solution and compare it to prior approximations. We also reveal connection between DTCM and q-deformed binomial statistics.

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Section: Probability Distribution Of Total Change Of Spin Polarimentioning

confidence: 99%

Landau-Zener extension of the Tavis-Cummings model: Structure of the solution

Sun

Sinitsyn

2016

Phys. Rev. A

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“…The limit q-Bernstein operator comes out as an analogue of the Szász-Mirakyan operator related to the Euler probability distribution, also called the -deformed Poisson distribution (see [1][2][3]). The latter is used in the -boson theory, which is a -deformation of the quantum harmonic oscillator formalism [4].…”

Section: Introductionmentioning

confidence: 99%

A Survey of Results on the Limit -Bernstein Operator

Ostrovska

2013

Journal of Applied Mathematics

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The limit -Bernstein operator emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution, which is used in the -boson theory to describe the energy distribution in a -analogue of the coherent state. At the same time, this operator bears a significant role in the approximation theory as an exemplary model for the study of the convergence of the -operators. Over the past years, the limit -Bernstein operator has been studied widely from different perspectives. It has been shown that is a positive shape-preserving linear operator on with . Its approximation properties, probabilistic interpretation, the behavior of iterates, and the impact on the smoothness of a function have already been examined. In this paper, we present a review of the results on the limit -Bernstein operator related to the approximation theory. A complete bibliography is supplied.

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Convergence properties of the q-deformed binomial distribution

Abstract: Abstract. We consider the q-deformed binomial distribution introduced by Jing 1994 and Chung et al. 1995 and establish several convergence results involving the Euler and the exponential distribution; some of them are qanalogues of classical results.

Cited by 2 publications

References 5 publications

Landau-Zener extension of the Tavis-Cummings model: Structure of the solution

Landau-Zener extension of the Tavis-Cummings model: Structure of the solution

A Survey of Results on the Limit -Bernstein Operator

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