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    Shannon entropy of symmetric Pollaczek polynomials

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    martinez-Shannon.pdf (328.0Kb)
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    URI: http://hdl.handle.net/10835/1635
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    Author/s
    Martínez-Finkelshtein, Andrei; Sánchez-Lara, J. F.
    Date
    2007
    Abstract
    We discuss the asymptotic behavior (as $n\to \infty$) of the entropic integrals $$ E_n= - \int_{-1}^1 \log \big(p^2_n(x) \big) p^2_n(x) w(x) d x, $$ and $$ F_n = -\int_{-1}^1 \log (p_n^2(x)w(x)) p_n^2(x) w(x) dx, $$ when $w$ is the symmetric Pollaczek weight on $[-1,1]$ with main parameter $\lambda\geq 1$, and $p_n$ is the corresponding orthonormal polynomial of degree $n$. It is well known that $w$ does not belong to the Szeg\H{o} class, which implies in particular that $E_n\to -\infty$. For this sequence we find the first two terms of the asymptotic expansion. Furthermore, we show that $F_n \to \log (\pi)-1$, proving that this ``universal behavior'' extends beyond the Szeg\H{o} class. The asymptotics of $E_n$ has also a curious interpretation in terms of the mutual energy of two relevant sequences of measures associated with $p_n$'s.
    Palabra/s clave
    Polinomios simétricos Pollaczek
    Entropía de Shannon
    Comportamiento asintótico
    Integrales entrópicas
    Shannon entropy
    Symmetric Pollaczek polynomials
    Asymptotic behavior
    Entropic integrals
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    • Artículos de revista Dpto. Matemáticas [119]

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