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The determination of the physical entropies (R\'enyi, Shannon, Tsallis) ofhigh-dimensional quantum systems subject to a central potential requires theknowledge of the asymptotics of some power and logarithmic integral functionalsof the hypergeometric orthogonal polynomials which control the wavefunctions ofthe stationary states. For the $D$-dimensional hydrogenic and oscillator-likesystems, the wavefunctions of the corresponding bound states are controlled bythe Laguerre ($\mathcal{L}_{m}^{(\alpha)}(x)$) and Gegenbauer($\mathcal{C}^{(\alpha)}_{m}(x)$) polynomials in both position and momentumspaces, where the parameter $\alpha$ linearly depends on $D$. In this work westudy the asymptotic behavior as $\alpha \to \infty$ of the associatedentropy-like integral functionals of these two families of hypergeometricpolynomials.

Entropic functionals of Laguerre and Gegenbauer polynomials with large parameters