Discrete Entropies of Orthogonal Polynomials

Aptekarev, A. I.; Dehesa, J. S.; Martínez–Finkelshtein, Andrei; Yáñez, R. J.

doi:10.1007/s00365-008-9028-9

Cited by 12 publications

(17 citation statements)

References 20 publications

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“…Remark 5. In [1, formula (27), p. 108], S n,1 is normalized in a different way because it lacks the normalizing factor 1/n, so that our formulas will slightly differ from those in [1].…”

Section: 1mentioning

confidence: 99%

“…By (28), the first term in (30) is 1 − 2 log(2) + R 1 2(2m+1) , while, by [1,Proposition 13,p. 114], we have the following identity for the second term…”

Section: 22mentioning

confidence: 99%

“…Thus, it is interesting to study the compatibility of the results from [1] with those stated in Theorem 1. In other words, can we reproduce (4), "stepping" onto the zeros ζ…”

mentioning

confidence: 95%

“…In what follows, when we have x = cos θ ∈ (−1, 1), then we also assume that θ ∈ (0, π). 1 We denote the set of nonnegative integers N ∪ {0} by N 0 .…”

mentioning

confidence: 99%

“…be the zeros of the n-th polynomial p n . In [1], the authors studied the values of S n,j = S(Ψ n (ζ (n) j )), j = 1, . .…”

mentioning

confidence: 99%

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Discrete entropy of generalized Jacobi polynomials

Martínez–Finkelshtein

Nevai

Peña

2015

Journal of Mathematical Analysis and Applications

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“…Remark 5. In [1, formula (27), p. 108], S n,1 is normalized in a different way because it lacks the normalizing factor 1/n, so that our formulas will slightly differ from those in [1].…”

Section: 1mentioning

confidence: 99%

“…By (28), the first term in (30) is 1 − 2 log(2) + R 1 2(2m+1) , while, by [1,Proposition 13,p. 114], we have the following identity for the second term…”

Section: 22mentioning

confidence: 99%

“…Thus, it is interesting to study the compatibility of the results from [1] with those stated in Theorem 1. In other words, can we reproduce (4), "stepping" onto the zeros ζ…”

mentioning

confidence: 95%

“…In what follows, when we have x = cos θ ∈ (−1, 1), then we also assume that θ ∈ (0, π). 1 We denote the set of nonnegative integers N ∪ {0} by N 0 .…”

mentioning

confidence: 99%

“…be the zeros of the n-th polynomial p n . In [1], the authors studied the values of S n,j = S(Ψ n (ζ (n) j )), j = 1, . .…”

mentioning

confidence: 99%

See 3 more Smart Citations

Discrete entropy of generalized Jacobi polynomials

Martínez–Finkelshtein

Nevai

Peña

2015

Journal of Mathematical Analysis and Applications

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Parameter andqasymptotics of Lq‐norms of hypergeometric orthogonal polynomials

Sobrino

Dehesa

2022

Int J of Quantum Chemistry

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The three canonical families of the hypergeometric orthogonal polynomials in a continuous real variable (Hermite, Laguerre, and Jacobi) control the physical wavefunctions of the bound stationary states of a great number of quantum systems [Correction added after first online publication on 21 December, 2022. The sentence has been modified.]. The algebraic frakturLq‐norms of these polynomials describe many chemical, physical, and information theoretical properties of these systems, such as, for example, the kinetic and Weizsäcker energies, the position and momentum expectation values, the Rényi and Shannon entropies and the Cramér‐Rao, the Fisher‐Shannon and LMC measures of complexity. In this work, we examine review and solve the q‐asymptotics and the parameter asymptotics (i.e., when the weight function's parameter tends towards infinity) of the unweighted and weighted frakturLq‐norms for these orthogonal polynomials. This study has been motivated by the application of these algebraic norms to the energetic, entropic, and complexity‐like properties of the highly excited Rydberg and high‐dimensional pseudo‐classical states of harmonic (oscillator‐like) and Coulomb (hydrogenic) systems, and other quantum systems subject to central potentials of anharmonic type (such as, e.g., some molecular systems) [Correction added after first online publication on 21 December, 2022. Oscillatorlike has been changed to oscillator‐like.].

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Information-Theoretic-Based Spreading Measures of Orthogonal Polynomials

Dehesa

Guerrero

Sánchez-Moreno

2011

Complex Anal. Oper. Theory

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The macroscopic properties of a quantum system strongly depend on the spreading of the physical eigenfunctions (wavefunctions) of its Hamiltonian operador over its confined domain. The wavefunctions are often controlled by classical or hypergeometric-type orthogonal polynomials (Hermite, Laguerre and Jacobi). Here we discuss the spreading of these polynomials over its orthogonality interval by means of various information-theoretic quantities which grasp some facets of the polynomial distribution not yet analyzed. We consider the information-theoretic lengths closely related to the Fisher information and Rényi and Shannon entropies, which quantify the polynomial spreading far beyond the celebrated standard deviation.

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Discrete Entropies of Orthogonal Polynomials

Abstract: Let p n , n ∈ N, be the nth orthonormal polynomial on R, whose zeros are λ

Cited by 12 publications

References 20 publications

Discrete entropy of generalized Jacobi polynomials

Discrete entropy of generalized Jacobi polynomials

Parameter andqasymptotics of Lq‐norms of hypergeometric orthogonal polynomials

Information-Theoretic-Based Spreading Measures of Orthogonal Polynomials

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