Narrow Results

The radial part of the wave function of an electron in a Coulomb potential is the product of a Laguerre polynomial and an exponential with the variable scaled by a factor depending on the degree. This note presents an elementary proof of the orthogonality of wave functions with differing energy levels. It is also shown that this is the only other natural orthogonality for Laguerre polynomials. By expanding in terms of the usual Laguerre polynomial basis, an analogous strange orthogonality is obtained for Meixner polynomials.

The generalized spectral-analytical method is based on the use of orthogonal decompositions of signals and analytic transformations in the space of expansion coefficients. The theory of classical orthogonal bases is a generalization of the theory of Fourier series to algebraic orthogonal polynomials of continuous and discrete arguments. The choice of the optimal set of features is carried out on the basis of integral signal estimates, and the mathematical operations on the signals studied are performed in the space of the expansion coefficients. Within the framework of the generalized spectral-analytical method, the problems of adaptive analytical description of one-dimensional and multidimensional digital information arrays are effectively solved in order to reduce the redundancy of signals’ description, as well as recognition problems on the basis of pairwise comparison of objects in Fourier coefficients features space.

This chapter describes the application of this method in the following problems:

• image recognition (by using invariants when recognizing contour objects),
• bioinformatics (recognition of repeats in sequences and complex spatial configurations of biomacromolecules),
• biomedicine (analysis of the spatial and temporal organization of electrical and magnetic activity of the brain).

The spreading of the four main families of classical orthogonal polynomials of a discrete variable (Hahn, Meixner, Kravchuk and Charlier), which are exact solutions of the second-order hypergeometric difference equation, is studied by means of some information-theoretic measures of global (variance, Shannon entropy power) and local (Fisher information) character. The variance is calculated in a closed an compact form by means of the three-term recurrence relation of the polynomials. Then, the Cramer-Rao and Heisenberg-Shannon inequalities are used to find rigorous bounds for the other two measures.

Meixner polynomials m_n(x; β, c) form a postive-definite orthogonal system on the positive real line x > 0 with respect to a distribution step function whose jumps are

Using the steepest descent method for oscillatory Riemann–Hilbert problems introduced by Deift and Zhou [Ann. Math.137 (1993), 295–368], we derive asymptotic formulas for the Meixner polynomials in two regions of the complex plane separated by the boundary of a rectangle. The asymptotic formula on the boundary of the rectangle is obtained by taking limits from either inside or outside. Our results agree with the ones obtained earlier for z on the positive real line by using the steepest descent method for integrals [Constr. Approx.14 (1998), 113–150].