Information-theoretic lengths of Jacobi polynomials

The information-theoretic lengths of the Jacobi polynomials P-n((alpha,beta)) (x), which are information-theoretic measures (Renyi, Shannon and Fisher) of their associated Rakhmanov probability density, are investigated. They quantify the spreading of the polynomials along the orthogonality interval [-1, 1] in a complementary but different way as the root-mean-square or standard deviation because, contrary to this measure, they do not refer to any specific point of the interval. The explicit expressions of the Fisher length are given. The Renyi lengths are found by the use of the combinatorial multivariable Bell polynomials in terms of the polynomial degree n and the parameters (alpha,beta). The Shannon length, which cannot be exactly calculated because of its logarithmic functional form, is bounded from below by using sharp upper bounds to general densities on [-1, + 1] given in terms of various expectation values; moreover, its asymptotics is also pointed out. Finally, several computational issues relative to these three quantities are carefully analyzed.