Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters

Abstract

The determination of the physical entropies (Rényi, Shannon, Tsallis) of high-dimensional quantum systems subject to a central potential requires the knowledge of the asymptotics of some power and logarithmic integral functionals of the hypergeometric orthogonal polynomials which control the wavefunctions of the stationary states. For the $D$-dimensional hydrogenic and oscillator-like systems, the wavefunctions of the corresponding bound states are controlled by the Laguerre (${\mathcal{L}}_m^{(\alpha )}(x)$) and Gegenbauer (${\mathcal{C}}_m^{(\alpha )}(x)$) polynomials in both position and momentum spaces, where the parameter $\alpha$ linearly depends on $D$. In this work we study the asymptotic behavior as $\alpha \to \infty$ of the associated entropy-like integral functionals of these two families of hypergeometric polynomials.

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Cited by in corpus (7)

• Entropic uncertainty measures for large dimensional hydrogenic systems
• Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems
• Analytical Shannon information entropies for all discrete multidimensional hydrogenic states
• Heisenberg and entropic uncertainty measures for large-dimensional harmonic systems
• Rényi entropies for multidimensional hydrogenic systems in position and momentum spaces
• Exact Rényi entropies of $D$-dimensional harmonic systems
• Complexity-like properties and parameter asymptotics of ${\mathfrak{L}}_q$-norms of Laguerre and Gegenbauer polynomials