Shannon entropy of symmetric Pollaczek polynomials

Abstract

We discuss the asymptotic behavior (as $n\to \infty$) of the entropic integrals $$E_n=-{\int }_{-1}^1\log (p_n^2(x))p_n^2(x)w(x)dx,$$ 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 $\lambda \ge 1$, and $p_n$ is the corresponding orthonormal polynomial of degree $n$. It is well known that $w$ does not belong to the Szegő 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 Szegő 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.

34 pages, 2 figures