A few remarks on orthogonal polynomials

Knowing a sequence of moments of a given, infinitely supported, distribution we obtain quickly: coefficients of the power series expansion of monic polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ that are orthogonal with respect to this distribution, coefficients of expansion of $x^{n}$ in the series of $p_{j},$ $j\leq n$, two sequences of coefficients of the 3-term recurrence of the family of $\left\{ p_{n}\right\} _{n\geq 0}$, the so called "linearization coefficients" i.e. coefficients of expansion of $$ in the series of $p_{j},$ $j\leq m+n.$\newline Conversely, assuming knowledge of the two sequences of coefficients of the 3-term recurrence of a given family of orthogonal polynomials $\left\{ p_{n}\right\} _{n\geq 0},$ we express with their help: coefficients of the power series expansion of $p_{n}$, coefficients of expansion of $x^{n}$ in the series of $p_{j},$ $j\leq n,$ moments of the distribution that makes polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ orthogonal. \newline Further having two different families of orthogonal polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ and $\left\{ q_{n}\right\} _{n\geq 0}$ and knowing for each of them sequences of the 3-term recurrences, we give sequence of the so called "connection coefficients" between these two families of polynomials. That is coefficients of the expansions of $p_{n}$ in the series of $q_{j},$ $j\leq n.$\newline We are able to do all this due to special approach in which we treat vector of orthogonal polynomials $\left\{ p_{j}\left( x)\right) \right\} _{j=0}^{n}$ as a linear transformation of the vector $\left\{ x^{j}\right\} _{j=0}^{n}$ by some lower triangular $(n+1)\times (n+1)$ matrix $\mathbf{\Pi }_{n}.$Comment: 18 page

Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation

We study the combinatorial structure of the irreducible characters of the classical groups ${\rm GL}_{n}(\mathbb{C})$, ${\rm SO}_{2n+1}(\mathbb{C})$, ${\rm Sp}_{2n}(\mathbb{C})$, ${\rm SO}_{2n}(\mathbb{C})$ and the "non-classical" odd symplectic group ${\rm Sp}_{2n+1}(\mathbb{C})$, finding new connections to the probabilistic model of Last Passage Percolation (LPP). Perturbing the expressions of these characters as generating functions of Gelfand-Tsetlin patterns, we produce two families of symmetric polynomials that interpolate between characters of ${\rm Sp}_{2n}(\mathbb{C})$ and ${\rm SO}_{2n+1}(\mathbb{C})$ and between characters of ${\rm SO}_{2n}(\mathbb{C})$ and ${\rm SO}_{2n+1}(\mathbb{C})$. We identify the first family as a one-parameter specialization of Koornwinder polynomials, for which we thus provide a novel combinatorial structure; on the other hand, the second family appears to be new. We next develop a method of Gelfand-Tsetlin pattern decomposition to establish identities between all these polynomials that, in the case of characters, can be viewed as describing the decomposition of irreducible representations of the groups when restricted to certain subgroups. Through these formulas we connect orthogonal and symplectic characters, and more generally the interpolating polynomials, to LPP models with various symmetries, thus going beyond the link with classical Schur polynomials originally found by Baik and Rains [BR01a]. Taking the scaling limit of the LPP models, we finally provide an explanation of why the Tracy-Widom GOE and GSE distributions from random matrix theory admit formulations in terms of both Fredholm determinants and Fredholm Pfaffians.Comment: 60 pages, 11 figures. Typos corrected and a few remarks adde

Fine Structure of the Zeros of Orthogonal Polynomials: A Review

We review recent work on zeros of orthogonal polynomials

Weak convergence of CD kernels and applications

We prove a general result on equality of the weak limits of the zero counting measure, $d\nu_n$, of orthogonal polynomials (defined by a measure $d\mu$) and $\frac{1}{n} K_n(x,x) d\mu(x)$. By combining this with Mate--Nevai and Totik upper bounds on $n\lambda_n(x)$, we prove some general results on $\int_I \frac{1}{n} K_n(x,x) d\mu_s\to 0$ for the singular part of $d\mu$ and $\int_I |\rho_E(x) - \frac{w(x)}{n} K_n(x,x)| dx\to 0$, where $\rho_E$ is the density of the equilibrium measure and $w(x)$ the density of $d\mu$

Asymptotics and zeros of Sobolev orthogonal polynomials on unbounded supports

In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we are focused in the study of the asymptotic behaviour of such polynomials as well as in the distribution of their zeros. Some open problems as well as some new directions for a future research are formulated.Comment: Changed content; 34 pages, 41 reference

Symmetric coupling of angular momenta, quadratic algebras and discrete polynomials

Eigenvalues and eigenfunctions of the volume operator, associated with the symmetric coupling of three SU(2) angular momentum operators, can be analyzed on the basis of a discrete Schroedinger-like equation which provides a semiclassical Hamiltonian picture of the evolution of a `quantum of space', as shown by the authors in a recent paper. Emphasis is given here to the formalization in terms of a quadratic symmetry algebra and its automorphism group. This view is related to the Askey scheme, the hierarchical structure which includes all hypergeometric polynomials of one (discrete or continuous) variable. Key tool for this comparative analysis is the duality operation defined on the generators of the quadratic algebra and suitably extended to the various families of overlap functions (generalized recoupling coefficients). These families, recognized as lying at the top level of the Askey scheme, are classified and a few limiting cases are addressed.Comment: 10 pages, talk given at "Physics and Mathematics of Nonlinear Phenomena" (PMNP2013), to appear in J. Phys. Conf. Serie