Information entropy of orthogonal polynomials

摘要

The Boltzmann–Shannon information entropy of quantum-mechanical systems in central potentials can be expressed in terms of the entropy En of the classical orthogonal polynomials,En=−∫p2n(x)logp2n(x)w(x)dx.Here, an explicit formula for the entropy of general orthogonal polynomials is obtained in terms of the eigenvalues of Jacobi matrix associated with the orthogonal polynomials pn(x) and the weighted moments of p2n(x),μn,k=∫xkp2n(x)w(x)dx,k=1,2,…