Shannon entropy of symmetric Pollaczek polynomials

We discuss the asymptotic behavior (as n→∞n→∞) of the entropic integralsEn=-∫-11logpn2(x)pn2(x)w(x)dx,andFn=-∫-11logpn2(x)w(x)pn2(x)w(x)dx,when ww is the symmetric Pollaczek weight on [-1,1][-1,1] with main parameter λ⩾1λ⩾1, and pnpn is the corresponding orthonormal polynomial of degree n  . It is well-known that ww does not belong to the Szegő class, which implies in particular that En→-∞En→-∞. For that sequence we find the first two terms of the asymptotic expansion. Furthermore, we show that Fn→log(π)-1Fn→log(π)-1, proving that this “universal behavior” extends beyond the Szegő class. The asymptotics of EnEn has also a curious interpretation in terms of the mutual energy of two relevant sequences of measures associated with pnpn's.