Shannon entropy of symmetric Pollaczek polynomials

We discuss the asymptotic behavior (as n → ∞) of the entropic integrals En = − ∫ 1 −1 log ( p n (x) ) p n (x)w(x) dx , and Fn = − ∫ 1 −1 log ( p n (x)w(x) ) p n (x)w(x) dx, when w is the symmetric Pollaczek weight on [−1, 1] with main parameter λ ≥ 1, and pn 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 En → −∞. For this sequence we find the first two terms of the asymptotic expansion. Furthermore, we show that Fn → log(π) − 1, proving that this “universal behavior” extends beyond the Szegő class. The asymptotics of En has also a curious interpretation in terms of the mutual energy of two relevant sequences of measures associated with pn’s.