Discrete entropies of orthogonal polynomials

Let pn, n ∈ N, be the nth orthonormal polynomial on R, whose zeros are λ j , j = 1, . . . , n. Then for each j = 1, . . . , n, ~ Ψj def = ( Ψ1j , . . . ,Ψ 2 nj ) with Ψij = p 2 i−1(λ (n) j ) ( n−1 ∑ k=0 pk(λ (n) j ) ) −1 , i = 1, . . . , n, defines a discrete probability distribution. The Shannon entropy of the sequence {pn} is consequently defined as Sn,j def = − n ∑ i=1 Ψij log ( Ψij ) . In the case of Chebyshev polynomials of the first and second kinds an explicit and closed formula for Sn,j is obtained, revealing interesting connections with the number theory. Besides, several results of numerical computations exemplifying the behavior of Sn,j for other families are also presented. AMS MOS Classification: 33C45, 41A58, 42C05, 94A17