Algebraic  and complexity‐like properties of Jacobi <scp>polynomials: Degree</scp> and parameter asymptotics

Sobrino, Nahual; Dehesa, J. S.

doi:10.1002/qua.26858

Cited by 3 publications

(4 citation statements)

References 70 publications

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“…which generalizes to all q the asymptotics given by (equation 35 of [101]) for the second-order norm W q P α,β ð Þ n h i of the orthogonal Jacobi polynomials. Moreover, the weighted L q -norms W q b P α,β ð Þ n of orthonormal Jacobi polynomials fulfill the parameter asymptotics…”

Section: Of Hops: Parameter Asymptoticsmentioning

confidence: 79%

“…To obtain the parameter asymptotics (

α \to \infty, β fixed

) of the unweighted norm

{scriptN}_{q} (P_{n}^{(α, β)})

and the Shannon entropy

E [{\overset{truê}{P}}_{n}^{(α, β)}]

of the Jacobi polynomials, given by Equations () and (), respectively, we follow the lines of Sobrino et al [101, section 3.2]. First, from Equation () and the limiting relation

\lim_{α \to \infty} \frac{P_{n}^{((),, α, β)} (x)}{P_{n}^{((),, α, β)} (1)} = {((), \frac{1 + x}{2})}^{n}, with P_{n}^{((),, α, β)} (1) = \frac{Γ (α + n + 1)}{n! Γ (α + 1)},

we find the asymptotics

{scriptN}_{p} [P_{n}^{(α, β)}] ~ \frac{Γ (α + n + 1)}{n!} \frac{Γ (1 + np + β)}{Γ (2 + α + np + β)} 2^{1 + α + β}; α \to \infty, β fixed

…”

Section: Frakturlq‐norms Scriptnqpn and Shannon Entropy Epn Of Hops: ...mentioning

confidence: 99%

“…So, let us center around the asymptotics (

λ \to \infty

) of the unweighted

{frakturL}_{q}

‐norms of orthogonal Gegenbauer polynomials given by

{scriptN}_{q} [C_{n}^{(λ)}] ≔ \int_{- 1}^{1} h_{λ}^{G} (x) {(||, C_{n}^{((), λ)})}^{q} dx,

and the Shannon entropy (), where

h_{λ}^{G} (x) = {((), 1 - x^{2})}^{λ - \frac{1}{2}}

. Then, we take into account the limiting relation

\lim_{λ \to \infty} \frac{C_{n}^{((), λ)} (x)}{C_{n}^{((), λ)} (1)} = x^{n}, with C_{n}^{((), λ)} (1) = \frac{(n + 2 λ - 1)!}{n! (2 λ - 1)!},

to obtain [101].

{scriptN}_{q} [C_{n}^{(λ)}] ~ {([], C_{n}^{(λ)} ((), 1))}^{q} \frac{Γ (\frac{1}{2} (1 + nq)) Γ (\frac{1}{2} + n)}{Γ (1 + λ + \frac{italicnq}{2})} ~

…”

Section: Frakturlq‐norms Scriptnqpn and Shannon Entropy Epn Of Hops: ...mentioning

confidence: 99%

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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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Section: Of Hops: Parameter Asymptoticsmentioning

confidence: 79%

“…To obtain the parameter asymptotics (

α \to \infty, β fixed

) of the unweighted norm

{scriptN}_{q} (P_{n}^{(α, β)})

and the Shannon entropy

E [{\overset{truê}{P}}_{n}^{(α, β)}]

of the Jacobi polynomials, given by Equations () and (), respectively, we follow the lines of Sobrino et al [101, section 3.2]. First, from Equation () and the limiting relation

\lim_{α \to \infty} \frac{P_{n}^{((),, α, β)} (x)}{P_{n}^{((),, α, β)} (1)} = {((), \frac{1 + x}{2})}^{n}, with P_{n}^{((),, α, β)} (1) = \frac{Γ (α + n + 1)}{n! Γ (α + 1)},

we find the asymptotics

{scriptN}_{p} [P_{n}^{(α, β)}] ~ \frac{Γ (α + n + 1)}{n!} \frac{Γ (1 + np + β)}{Γ (2 + α + np + β)} 2^{1 + α + β}; α \to \infty, β fixed

…”

Section: Frakturlq‐norms Scriptnqpn and Shannon Entropy Epn Of Hops: ...mentioning

confidence: 99%

“…So, let us center around the asymptotics (

λ \to \infty

) of the unweighted

{frakturL}_{q}

‐norms of orthogonal Gegenbauer polynomials given by

{scriptN}_{q} [C_{n}^{(λ)}] ≔ \int_{- 1}^{1} h_{λ}^{G} (x) {(||, C_{n}^{((), λ)})}^{q} dx,

and the Shannon entropy (), where

h_{λ}^{G} (x) = {((), 1 - x^{2})}^{λ - \frac{1}{2}}

. Then, we take into account the limiting relation

\lim_{λ \to \infty} \frac{C_{n}^{((), λ)} (x)}{C_{n}^{((), λ)} (1)} = x^{n}, with C_{n}^{((), λ)} (1) = \frac{(n + 2 λ - 1)!}{n! (2 λ - 1)!},

to obtain [101].

{scriptN}_{q} [C_{n}^{(λ)}] ~ {([], C_{n}^{(λ)} ((), 1))}^{q} \frac{Γ (\frac{1}{2} (1 + nq)) Γ (\frac{1}{2} + n)}{Γ (1 + λ + \frac{italicnq}{2})} ~

…”

Section: Frakturlq‐norms Scriptnqpn and Shannon Entropy Epn Of Hops: ...mentioning

confidence: 99%

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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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“…This is because the necessary experiments are less accessible and the required computational work is much bigger. Nevertheless, the disorder quantifiers and uncertainty measures have been recently determined because the behavior of the involved integral functionals of the hydrogenic-type wavefunctions at Rydberg limit n → ∞ have been analytically estimated by use of fine asymptotics of orthogonal polynomials (Laguerre, Gegenbauer) which, together with the hyperspherical harmonics [18][19][20][21][22], describe the state's wavefunctions.…”

Section: Introductionmentioning

confidence: 99%

Rydberg atoms in D dimensions: entanglement, entropy and complexity

Dehesa

2024

J. Phys. A: Math. Theor.

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Rydberg atoms and excitons are composed so that they have a hydrogenic energy level structure governed by the Rydberg formula. They are relevant per se and for their numerous applications, e.g. facilitating the creation of novel quantum devices in quantum technologies which are inherently robust, miniature, and scalable (basically because they exist in solid-state platforms) and the realization of synthetic dimensions in numerous quantum-mechanical systems, giving rise to quantum matter which can behave as if it were in dimensions other than three. However the quantification of their internal disorder is scarcely known. Here we show and review the knowledge of dispersion, entanglement, physical entropies (Rényi, Shannon) and complexity-like measures of D-dimensional Rydberg systems with $D \geq 2$ in both position and momentum spaces. These uncertainty quantifiers are expressed in terms of D, the potential strength and the hyperquantum numbers of the Rydberg states. This has been possible because of the fine asymptotics of algebraic functionals the Laguerre and Gegenbauer polynomials which, together with the hyperspherical harmonics, control the Rydberg wavefunctions.

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Cramér–Rao, Fisher–Shannon and LMC–Rényi Complexity-like Measures of Multidimensional Hydrogenic Systems with Application to Rydberg States

Dehesa

2023

Quantum Reports

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Statistical measures of complexity hold significant potential for applications in D-dimensional finite fermion systems, spanning from the quantification of the internal disorder of atoms and molecules to the information–theoretical analysis of chemical reactions. This potential will be shown in hydrogenic systems by means of the monotone complexity measures of Cramér–Rao, Fisher–Shannon and LMC(Lopez-Ruiz, Mancini, Calbet)–Rényi types. These quantities are shown to be analytically determined from first principles, i.e., explicitly in terms of the space dimensionality D, the nuclear charge and the hyperquantum numbers, which characterize the system’ states. Then, they are applied to several relevant classes of particular states with emphasis on the quasi-spherical and the highly excited Rydberg states, obtaining compact and physically transparent expressions. This is possible because of the use of powerful techniques of approximation theory and orthogonal polynomials, asymptotics and generalized hypergeometric functions.

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Algebraic and complexity‐like properties of Jacobi polynomials: Degree and parameter asymptotics

Cited by 3 publications

References 70 publications

Parameter andqasymptotics of Lq‐norms of hypergeometric orthogonal polynomials

Parameter andqasymptotics of Lq‐norms of hypergeometric orthogonal polynomials

Rydberg atoms in D dimensions: entanglement, entropy and complexity

Cramér–Rao, Fisher–Shannon and LMC–Rényi Complexity-like Measures of Multidimensional Hydrogenic Systems with Application to Rydberg States

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