Multidimensional bell polynomials of higher order

In this study, the information-theoretic measures in both the position and momentum spaces for the pseudoharmonic potential using Fisher information, Shannon entropy, Renyi entropy, Tsallis entropy and Onicescu information energy are investigated analytically and numerically. The results obtained are applied to some diatomic molecules. The Renyi and Tsallis entropies are analytically obtained in position space using Srivastava-Niukkanen linearization formula in terms of the Lauricella hypergeometric function. Also they are obtained in the momentum space in terms of the multivariate Bell polynomials of Combinatorics. We observed that the Fisher information increases with n in both the position and momentum spaces, but decreases with ℓ for all the diatomic molecules considered. The Shannon entropy also increases with increasing n in the position space and decreases with increasing ℓ. The variations of the Renyi and Tsallis entropies with ℓ are also discussed. The exact and numerical values of the Onicescu information energy are also obtained, after which the ratio of information-theoretic impetuses to lengths for Fisher, Shannon and Renyi are obtained.

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“…. The Bell polynomials are given by

B_{m, t} (a_{1}, a_{2}, ..., a_{m - t + 1}) = \sum_{\overset{π}{true} false(m, t false)} \frac{m!}{j_{1}! j_{2}! ... j_{m - t + 1}!} {true(\frac{a_{1}}{1!} true)}^{j_{1}} {true(\frac{a_{2}}{2!} true)}^{j_{2}} ... {true(\frac{a_{m - t + 1}}{(m - t + 1)!} true)}^{j_{m - t + 1}},

where the sum runs over all partitions

\hat{π} (m, t)

such that

j_{1} + j_{2} + ... + j_{m - t + 1} = t, and j_{1} + 2 j_{2} + ... + (m - t + 1) j_{m - t + 1} = m .

Therefore,

H_{q} [γ] = \sum_{k = 0}^{2 n q} \frac{(2 q)!}{(k + 2 q)!} B_{k + 2 q, 2 q} true(c_{0}^{(n, γ_{ℓ} + \frac{1}{2}}

…”

Section: Renyi Entropy Tsallis Entropy and Onicescu Information Enementioning

confidence: 99%

Position and momentum information‐theoretic measures of the pseudoharmonic potential

Yahya

Oyewumi

Sen

2015

Int J of Quantum Chemistry

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“…So that they can be related to two dimensionless bosonic fields v and u : R 2,1 → 0 , by setting with c 1 , c 2 ∈ 0 being free function to be appropriate choices such that the Equation (19) connects with a linear combination system of super binary Bell polynomials. Substituting (21) into (19) and integrating with respective to D and x, respectively, yields…”

Section: Bilinear Representationmentioning

confidence: 99%

“…The Bell polynomials have been exploited in combinatorics, statistics, and other fields [15][16][17]. Some generalized forms of Bell polynomials have already appeared in literature [18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

New Bilinear Bäcklund Transformation and Lax Pair for the Supersymmetric Two‐Boson Equation

Fan

2011

Stud Appl Math

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In this paper, I introduce a class of super Bell polynomials, which are found to play an important role in the characterization of super supersymmetric equations. An effective approach based on the use of the super Bell polynomials is developed to systematically investigate the bilinearization, Bäcklund transformation, and Lax pair for supersymmetric equations. I take a supersymmetric two‐boson equation to illustrate this procedure. A new bilinear Bäcklund transformation and a Lax pair with both fermionic and bosonic parameters are given. In addition, a kind of exact solitons for the equation are further constructed with the help of the bilinear Bäcklund transformation.

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“…[14][15][16][17]). A generalization of the Bell polynomials suitable for the differentiation of multivariable composite functions can also be found in [18,19].…”

Section: Proposition 21 the Bell Polynomials Satisfy The Recurrencementioning

confidence: 99%

Bell polynomials and modified Bessel functions of half-integral order

Natalini

Ricci

2015

Applied Mathematics and Computation

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Multidimensional bell polynomials of higher order

Cited by 28 publications

References 6 publications

Position and momentum information‐theoretic measures of the pseudoharmonic potential

Position and momentum information‐theoretic measures of the pseudoharmonic potential

New Bilinear Bäcklund Transformation and Lax Pair for the Supersymmetric Two‐Boson Equation

Bell polynomials and modified Bessel functions of half-integral order

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