Related papers: Mixed Multiple Orthogonal Laurent Polynomials on t…
Recursive algebraic construction of two infinite families of polynomials in $n$ variables is proposed as a uniform method applicable to every semisimple Lie group of rank $n$. Its result recognizes Chebyshev polynomials of the first and…
We obtain representations of $X_1$ exceptional orthogonal polynomials through determinants of matrices that have certain adjusted moments as entries. We start out directly from the Darboux transformation, allowing for a universal…
It is shown that the CMV Laurent polynomials associated to the sieved Jacobi polynomials on the unit circle satisfy an eigenvalue equation with respect to a first order differential operator of Dunkl type. Using this result, the sieved…
A set of orthogonal polynomials on the unit disk $B(0,1)$ known as Zernike polynomials are commonly used in the analysis and evaluation of optical systems. Here Zernike polynomials are used to construct wavelets for polynomial subspaces of…
In this paper we study matrix valued orthogonal polynomials of one variable associated with a compact connected Gelfand pair (G,K) of rank one, as a generalization of earlier work by Koornwinder and subsequently by Koelink, van Pruijssen…
Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Q_n(x,y};N,p) on the multinomial distribution…
Brion's Formula realizes the Laurent polynomial of lattice points in a lattice polytope P as the sum of rational functions associated to the vertices of P. In this paper, we consider the special case where P is a generalized permutohedron.…
The numerical complex coupled-mode method used in a metal thin-film optic element is applied to a planar multilayer optical waveguide. All modes are required to satisfy Helmholtz Vectorial equation in an optical waveguide including bound…
First we argue that many BV and homotopy BV structures, including both familiar and new examples, arise from a common underlying construction. The input of this construction is a cyclic operad along with a cyclically invariant Maurer-Cartan…
This paper is devoted to characterizations classical orthogonal polynomials on quadratic lattices by using a matrix approach. In this form we recover the Hahn, Geronimus, Tricomi and Bochner type characterizations of classical orthogonal…
We introduce and study a special family of polynomials orthogonal on the unit circle (OPUC). These OPUC satisfy a mirror symmetry property of their Verblunsky coefficients. Several equivalent conditions for the OPUC to be mirror symmetric…
We provide a detailed treatment of Weyl-Titchmarsh theory for half-lattice and full-lattice Cantero-Moral-Velazquez (CMV) operators and discuss their systems of orthonormal Laurent polynomials on the unit circle, spectral functions,…
The orthogonal polynomials with recurrence relation \[(\la\_n+\mu\_n-z) F\_n(z)=\mu\_{n+1} F\_{n+1}(z)+\la\_{n-1} F\_{n-1}(z)\] with two kinds of cubic transition rates $\la\_n$ and $\mu\_n,$ corresponding to indeterminate Stieltjes moment…
Let $\un,$ be a quasi-definite linear functional defined on the space of polynomials $\mathbb{P}.$ For such a functional we can define a sequence of monic orthogonal polynomials (SMOP in short) $(P_n)_{n\geq 0},$ which satisfies a three…
We give four examples of families of orthogonal polynomials for which the coefficients in the recurrence relation satisfy a discrete Painlev\'e equation. The first example deals with Freud weights $|x|^\rho \exp(-|x|^m)$ on the real line,…
The matrix integral has many applications in diverse fields. This review article begins by presenting detailed key background knowledge about matrix integral. Then the volumes of orthogonal groups and unitary groups are computed,…
Dual Bernstein polynomials find many applications in approximation theory, computational mathematics, numerical analysis and computer-aided geometric design. In this context, one of the main problems is fast and accurate evaluation both of…
In this paper, we undertake a systematic study of recurrences x_{m+n}x_{m} = P(x_{m+1}, ..., x_{m+n-1}) which exhibit the Laurent phenomenon. Some of the most famous among these sequences come from the Somos and the Gale-Robinson…
Using a lemma of Davis on Gram matrices applied to the classical Orthogonal Polynomials to generate reproducing kernel interpolation over the classical domains for polynomials. These kernels have terms which are exact over the rational…
This work explores the tensor and combinatorial constructs underlying the linearised higher-order variational equations of a generic autonomous system along a particular solution. The main result of this paper is a compact yet explicit and…