Related papers: The Askey scheme as a four-manifold with corners
In this paper we consider the automorphisms of the double affine Hecke algebra (DAHA) of type $\check{C_1}C_1$ which have a relatively simple action on the generators and on the parameters, notably a symmetry $t_4$ which sends the…
The double-direction orthogonalization algorithm is applied to construct sequences of polynomials, which are orthogonal over the interval [0,1]with the weighting function 1. Functional and recurrent relations are derived for the sequences…
In this paper we describe the well studied process of renormalization of quadratic polynomials from the point of view of their natural extensions. In particular, we describe the topology of the inverse limit of infinitely renormalizable…
The geometrical diffraction theory, in the sense of Keller,is here reconsidered as an obstacle problem in the Riemannian geometry. The first result is the proof of the existence and the analysis of the main properties of the diffracted…
We describe a polynomial time algorithm that takes as input a polygon with axis-parallel sides but irrational vertex coordinates, and outputs a set of as few rectangles as possible into which it can be dissected by axis-parallel cuts and…
Rigidity results are obtained for Riemannian $d$-manifolds with $\sec \geqslant 1$ and spherical rank at least $d-2>0$. Conjecturally, all such manifolds are locally isometric to a round sphere or complex projective space with the…
We establish, for every family of orthogonal polynomials in the $ q $-Askey scheme and the Askey scheme, a combinatorial model for mixed moments and coefficients in terms of paths on the lecture hall graph. This generalizes the previous…
In this chapter we characterize Askey-Wilson polynomials including specific and limiting cases of them by some structure relations of the first type.
We study a rational version of the double affine Hecke algebra associated to the nonreduced affine root system of type $(C^\vee_n,C_n)$. A certain representation in terms of difference-reflection operators naturally leads to the definition…
We present a one-parameter family of constant solutions of the reflection equation and define a family of quantum complex Grassmannians endowed with a transitive action of the quantum unitary group. By computing the radial part of a…
In this paper, we study staircase tableaux, a combinatorial object introduced due to its connections with the asymmetric exclusion process (ASEP) and Askey-Wilson polynomials. Due to their interesting connections, staircase tableaux have…
The Al-Salam-Chihara polynomials are an important family of orthogonal polynomials in one variable $x$ depending on 3 parameters $\alpha$, $\beta$ and $q$. They are closely connected to a model from statistical mechanics called the…
The interpretation of the Meixner-Pollaczek, Meixner and Laguerre polynomials as overlap coefficients in the positive discrete series representations of the Lie algebra su(1,1) and the Clebsch-Gordan decomposition leads to generalisations…
In this paper, a fourth-order partial divided-difference equation on quadratic lattices with polynomial coefficients satisfied by bivariate Racah polynomials is presented. From this equation we obtain explicitly the matrix coefficients…
In this survey paper, we exhaustively explore the terminating basic hypergeometric representations of the Askey-Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy. From…
A parametric manifold can be viewed as the manifold of orbits of a (regular) foliation of a manifold by means of a family of curves. If the foliation is hypersurface orthogonal, the parametric manifold is equivalent to the 1-parameter…
This article is concerned with a geometric tool given by a pair of projector operators defined by almost product structures on finite dimensional manifolds, polarized by a distribution of constant rank and also endowed with some geometric…
We characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. Some applications are given to complex manifold and to strongly minimal sets.
The standard theory of stochastic approximation (SA) is extended to the case when the constraint set is a Riemannian manifold. Specifically, the standard ODE method for analyzing SA schemes is extended to iterations constrained to stay on a…
Systems of orthogonal polynomials whose recurrence coefficients tend to infinity are considered. A summability condition is imposed on the coefficients and the consequences for the measure of orthogonality are discussed. Also discussed are…