Formulas For Orthogonal Polynomials Research Articles

Wall Polynomials on the Real Line: A Classical Approach to OPRL Khrushchev’s Formula

The standard proof of Khrushchev’s formula for orthogonal polynomials on the unit circle given in Khrushchev (J Approx Theory 108:161–248, 2001, J Approx Theory 116:268–342, 2002) combines ideas from continued fractions and complex analysis, depending heavily on the theory of Wall polynomials. Using operator theoretic tools instead, Khrushchev’s formula has been recently extended to the setting of orthogonal polynomials on the real line in the determinate case (Grünbaum and Velázquez in Adv Math 326:352–464, 2018). This paper develops a theory of Wall polynomials on the real line, which serves as a means to prove Khrushchev’s formula for any sequence of orthogonal polynomials on the real line. This real line version of Khrushchev’s formula is used to rederive the characterization given in Simon (J Approx Theory 126:198–217, 2004) for the weak convergence of p_n^2mathrm{d}mu , where p_n are the orthonormal polynomials with respect to a measure mu supported on a bounded subset of the real line (Theorem 8.1). The generality and simplicity of such a Khrushchev’s formula also permits the analysis of the unbounded case. Among other results, we use this tool to prove that no measure mu supported on an unbounded subset of the real line yields a weakly convergent sequence p_n^2mathrm{d}mu (Corollary 8.10), but there exist instances such that p_n^2mathrm{d}mu becomes vaguely convergent (Example 8.5 and Theorem 8.6). Some other asymptoptic results related to the convergence of p_n^2mathrm{d}mu in the unbounded case are obtained via Khrushchev’s formula (Theorems 8.3, 8.7, 8.8, Proposition 8.4, Corollary 8.9). In the bounded case, we include a simple diagrammatic proof of Khrushchev’s formula on the real line which sheds light on its graph theoretical meaning, linked to Pólya’s recurrence theory for classical random walks.

Poincaré series of Klein groups, coxeter polynomials, the Burau representation, and Milnor invariants

We obtain several formulas for the Poincare series defined by B. Kostant for Klein groups (binary polyhedral groups) and some formulas for Coxeter polynomials (characteristic polynomials of monodromy in the case of singularities). Some of these formulas—the generalized Ebeling formula, the Christoffel-Darboux identity, and the combinatorial formula—are corollaries to the well-known statements on the characteristic polynomial of a graph and are analogous to formulas for orthogonal polynomials. The ratios of Poincare series and Coxeter polynomials are represented in terms of branched continued fractions, which are q-analogs of continued fractions that arise in the theory of resolution of singularities and in the Kirby calculus. Other formulas connect the ratios of some Poincare series and Coxeter polynomials with the Burau representation and Milnor invariants of string links. The results obtained by S.M. Gusein-Zade, F. Delgado, and A. Campillo allow one to consider these facts as statements on the Poincare series of the rings of functions on the singularities of curves, which suggests the following conjecture: the ratio of the Poincare series of the rings of functions for close (in the sense of adjacency or position in a series) singularities of curves is determined by the Burau representation or by the Milnor invariants of a string link, which is an intermediate object in the transformation of the knot of one singularity into the knot of the other.

Continued fractions and orthogonal polynomials on the unit circle

This survey is written to stress the role of continued fractions in the theory of orthogonal polynomials on the line and on the circle. We follow the historical development of the subject, which opens many interesting relationships of orthogonal polynomials to other important branches of mathematics. At the end we present a new formula for orthogonal polynomials on the real line, the Leganés formula, ∫ Q n - 1 2 d σ t - z = 1 Q n / Q n - 1 - ∫ d σ n / ( t - z ) , which is a correct analogue of the corresponding formula on the unit circle. This formula is applied to obtain a recent result by Simon.

Uniform Asymptotic Formula for Orthogonal Polynomials with Exponential Weight

Let {pn(x)}_{n\ge 0}$ be the set of orthonormal polynomials with respect to the exponential weight w(x)=e-v(x) , where v(x)=x2m + ... is a monic polynomial of degree 2m with $m \ge 2$ and is even. An asymptotic approximation is obtained for pn(x), as $n \to \infty$, which holds uniformly for $0 \le x \le O(n^{1/2m})$. As a corollary, a three-term asymptotic expansion is also derived for the zeros of these polynomials.

An analog of the trace formula for orthogonal polynomials with asymptoticallyN-periodic recurrence coefficients (N=2)

An analog of the trace formula for orthogonal polynomials with asymptotically 2-periodic recurrence coefficients is given. This result is based on the representation of the Fejer kernel for the system.

A new presentation of orthogonal polynomials with applications to their computation

In this paper a new presentation of orthogonal polynomials is given. It is based on the introduction of two auxiliary sequences of arbitrary monic polynomials and it leads to a very simple derivation of the usual determinantal formulae for orthogonal polynomials and of their recurrence relations either in the definite or in the indefinite case. New expressions for the coefficients of these recurrence relations are obtained and they are compared to the usual ones from the point of view of their numerical stability. The qd-algorithm is also recovered very easily.

Orthogonal Polynomials Whose Distribution Functions Have Finite Point Spectra

A counterexample is given to an assertion by K. M. Case that if the coefficients in the three-term recurrence formula for orthogonal polynomials converge as fast as $n^{ - 2} $, the corresponding distribution function has only finitely many discrete points in its spectrum. Some positive results concerning this situation are also given, and continuity of the distribution function is investigated.