Shannon entropy of symmetric Pollaczek polynomials

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.