Narrow Results

In this paper, we develop the Riemann–Hilbert approach to study the global asymptotics of discrete orthogonal polynomials with infinite nodes. We illustrate our method by concentrating on the Charlier polynomials . We first construct a Riemann–Hilbert problem Y associated with these polynomials and then establish some technical results to transform Y into a continuous Riemann–Hilbert problem so that the steepest descent method of Deift and Zhou ([8]) can be applied. Finally, we produce three Airy-type asymptotic expansions for in three different but overlapping regions whose union is the entire complex z-plane. When z is real, our results agree with the ones given in the literature. Although our approach is similar to that used by Baik, Kriecherbauer, McLaughlin and Miller ([3]), there are crucial differences in the details. For instance, our expansions hold in much bigger regions. Our results are completely new, and one of them answers a question raised in Bo and Wong ([4]). Asymptotic formulas are also derived for large and small zeros of the Charlier polynomials.

We derive uniform and non-uniform asymptotics of the Charlier polynomials by using difference equation methods alone. The Charlier polynomials are special in that they do not fit into the framework of the turning point theory, despite the fact that they are crucial in the Askey scheme. In this paper, asymptotic approximations are obtained, respectively, in the outside region, an intermediate region, and near the turning points. In particular, we obtain uniform asymptotic approximation at a pair of coalescing turning points with the aid of a local transformation. We also give a uniform approximation at the origin by applying the method of dominant balance and several matching techniques.

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).

In this paper, we develop the Riemann–Hilbert approach to study the global asymptotics of discrete orthogonal polynomials with infinite nodes. We illustrate our method by concentrating on the Charlier polynomials . We first construct a Riemann–Hilbert problem Y associated with these polynomials and then establish some technical results to transform Y into a continuous Riemann–Hilbert problem so that the steepest descent method of Deift and Zhou can be applied. Finally, we produce three Airy-type asymptotic expansions for in three different but overlapping regions whose union is the entire complex z-plane. When z is real, our results agree with the ones given in the literature. Although our approach is similar to that used by Baik, Kriecherbauer, McLaughlin and Miller, there are crucial differences in the details. For instance, our expansions hold in much bigger regions. Our results are completely new, and one of them answers a question raised in Bo and Wong. Asymptotic formulas are also derived for large and small zeros of the Charlier polynomials.