Fisher information of special functions and second-order differential equations

Abstract

We investigate a basic question of information theory, namely the evaluation of the Fisher information and the relative Fisher information with respect to a nonnegative function, for the probability distributions obtained by squaring the special functions of mathematical physics which are solutions of second-order differential equations. Emphasis is made in the Nikiforov-Uvarov hypergeometric-type functions. We obtain explicit expressions for these information-theoretic properties via the expectation values of the coefficients of the differential equation. We illustrate our approach for various special functions of physico-mathematical interest.

On minimum Fisher information distributions with restricted support and fixed variance

“…In robust estimation, minimization of Fisher information has been originally considered by Huber [15], and in the case of scale and location parameters in a Kolmogorov neighborhood of a parent distribution in [33,34]. Fisher information for orthogonal polynomials and special functions have been studied in [24,36]. Connections with the differential equations of Physics have been explored in [11].…”

Information entropy of conditionally exactly solvable potentials

“…In view of this we feel it would be interesting to study asymptotic behaviour (with respect to the quantum number) of the entropies of exceptional orthogonal polynomials. Finally we would like to mention that it would be worth investigating some other properties like the Fisher information [23], spreading length [24] Table 2. Information Entropies for the eigenstates of the potentials in (10) and (12) for l = m = 0.…”

Hellmann–Feynman connection for the relative Fisher information

“…From (72), it is evident that next three terms in the LHS of (74) comprise λ RFI 0 , and the expectations are specified as: (11) and (72) allows for the reconstruction of the RFI framework from the FIM framework.…”

“…From (11), the LHS of (75) is I[ψ 2 ], where again it is deliberately unspecified as to whether ψ extremizes the RFI or the FIM. However, from (72) it is evident that first three terms in the RHS of (75) comprise λ FIM 0 .…”