Relative Fisher information of discrete classical orthogonal polynomials

TL;DR

This paper extends the analytic information theory of discrete distributions by deriving explicit expressions for the relative Fisher information of Rakhmanov distributions associated with classical discrete orthogonal polynomials.

Contribution

It introduces a broader class of distributions and provides explicit formulas for their relative Fisher information, advancing the understanding of information measures in discrete orthogonal polynomials.

Findings

Explicit formulas for relative Fisher information of Charlier, Meixner, Kravchuk, Hahn distributions.

Extension of previous work to a larger class of Rakhmanov distributions.

Enhanced understanding of information measures in discrete orthogonal polynomial families.

Abstract

The analytic information theory of discrete distributions was initiated in 1998 by C. Knessl, P. Jacquet and S. Szpankowski who addressed the precise evaluation of the Renyi and Shannon entropies of the Poisson, Pascal (or negative binomial) and binomial distributions. They were able to derive various asymptotic approximations and, at times, lower and upper bounds for these quantities. Here we extend these investigations in a twofold way. First, we consider a much larger class of distributions, the Rakhmanov distributions $\rho_n(x)=\omega(x)y_n^2(x)$, where $\{y_n(x)\}$ denote the sequences of discrete hypergeometric-type polynomials which are orthogonal with respect to the weight function $\omega(x)$ of Poisson, Pascal, binomial and hypergeometric types; that is the polynomials of Charlier, Meixner, Kravchuk and Hahn. Second, we obtain the explicit expressions for the relative Fisher information of these four families of Rakhmanov distributions with respect to their respective weight functions.