On semi-classical weight functions on the unit circle

We consider orthogonal polynomials on the unit circle associated with certain semi-classical weight functions. This means that the Pearson-type differential equations satisfied by these weight functions involve two polynomials of degree at most 2. We determine all such semi-classical weight functions and this also includes an extension of the Jacobi weight function on the unit circle. General structure relations for the orthogonal polynomials and non-linear difference equations for the associated complex Verblunsky coefficients are established. As application, we present several new structure relations and non-linear difference equations associated with some of these semi-classical weight functions.Comment: 26 page

On polynomial solutions of differential equations

A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the "projectivized" representation possessing an invariant subspace and the spectral problem for a certain linear differential operator with variable coefficients. It is shown in general that polynomial solutions of partial differential equations occur; in the case of Lie superalgebras there are polynomial solutions of some matrix differential equations, quantum algebras give rise to polynomial solutions of finite--difference equations. Particularly, known classical orthogonal polynomials will appear when considering $SL(2,{\bf R})$ acting on ${\bf RP_1}$. As examples, some polynomials connected to projectivized representations of $sl_2 ({\bf R})$, $sl_2 ({\bf R})_q$, $osp(2,2)$ and $so_3$ are briefly discussed.Comment: 12p

Hypergeometric multiple orthogonal polynomials

This thesis is devoted to the analysis of multiple orthogonal polynomials for indices on the so-called step-line with respect to absolutely continuous measures on the positive real line, whose moments are given by ratios of Pochhammer symbols (also known as rising factorials). We investigate both type I and type II multiple orthogonal polynomials, though the main focus is on the type II polynomials. For the former, the characterisation includes Rodrigues-type formulas for the type I polynomials and type I functions. On the latter, the characterisation includes explicit representations as terminating generalised hypergeometric series as well as solutions of differential equations and recurrence relations, and an analysis of their asymptotic behaviour and the location of their zeros. We investigate the link of these polynomials with branched-continued-fraction representations of generalised hypergeometric series, which were introduced to solve total-positivity problems in combinatorics. The polynomials analysed here also have direct applications to the study of Painlevé equations and to random matrix theory. We give a detailed characterisation of two new families of multiple orthogonal polynomials associated with Nikishin systems of 2 absolutely continuous measures. These measures are supported on the positive real line and on the interval (0,1) and they admit integral representations via the confluent hypergeometric function of the second kind (also known as the Tricomi function) and Gauss' hypergeometric function, respectively. The vectors of orthogonality weights satisfy matrix Pearson-type differential equations, linked to the action of the differentiation operator on the type II polynomials and type I functions as a shift in their index and parameters. As a result, the type II polynomials and type I functions satisfy Hahn's property. We further draw the links between these two families of multiple orthogonal polynomials and other known polynomial sets via limiting relations or specialisations. Examples of such connections encompass the components of the cubic decomposition of Hahn-classical threefold-symmetric 2-orthogonal polynomials as well as Jacobi-Piñeiro polynomials and multiple orthogonal polynomials with respect to Macdonald functions

Using \D-operators to construct orthogonal polynomials satisfying higher order difference or differential equations

We introduce the concept of \D-operators associated to a sequence of polynomials $(p_n)_n$ and an algebra \A of operators acting in the linear space of polynomials. In this paper, we show that this concept is a powerful tool to generate families of orthogonal polynomials which are eigenfunctions of a higher order difference or differential operator. Indeed, given a classical discrete family $(p_n)_n$ of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we form a new sequence of polynomials $(q_n)_n$ by considering a linear combination of two consecutive $p_n$: $q_n=p_n+\beta_np_{n-1}$, \beta_n\in \RR. Using the concept of \D-operator, we determine the structure of the sequence $(\beta_n)_n$ in order that the polynomials $(q_n)_n$ are common eigenfunctions of a higher order difference operator. In addition, we generate sequences $(\beta_n)_n$ for which the polynomials $(q_n)_n$ are also orthogonal with respect to a measure. The same approach is applied to the classical families of Laguerre and Jacobi polynomials.Comment: 43 page

Semi-classical Laguerre polynomials and a third order discrete integrable equation

A semi-discrete Lax pair formed from the differential system and recurrence relation for semi-classical orthogonal polynomials, leads to a discrete integrable equation for a specific semi-classical orthogonal polynomial weight. The main example we use is a semi-classical Laguerre weight to derive a third order difference equation with a corresponding Lax pair.Comment: 11 page