Related papers: Orthogonal and multiple orthogonal polynomials, ra…
In this paper we establish a connection between the fluctuations of Wishart random matrices, shifted Chebyshev polynomials, and planar diagrams whose linear span form a basis for the irreducible representations of the annular Temperly-Lieb…
We provide two new characterizations of bounded orthogonally additive polynomials from a uniformly complete vector lattice into a convex bornological space using separately two polynomial identities of Kusraeva involving the root mean power…
We characterize the atomic probability measure on $\mathbb{R}^d$ which having a finite number of atoms. We further prove that the Jacobi sequences associated to the multiple Hermite (resp. Laguerre, resp. Jacobi) orthogonal polynomials are…
We characterize the biorthogonal polynomials that appear in the theory of coupled random matrices via a Riemann-Hilbert problem. Our Riemann-Hilbert problem is different from the ones that were proposed recently by Ercolani and McLaughlin,…
Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…
Let $(P_n)_n$ and $(Q_n)_n$ be two sequences of monic polynomials linked by a type structure relation such as $$ Q_{n}(x)+r_nQ_{n-1}(x)=P_{n}(x)+s_nP_{n-1}(x)+t_nP_{n-2}(x)\;, $$ where $(r_n)_n$, $(s_n)_n$ and $(t_n)_n$ are sequences of…
We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary co-adjoint orbits of the Borel group of upper triangular matrices; these orbits are the phase space of…
We formulate and prove a general recurrence relation that applies to integrals involving orthogonal polynomials and similar functions. A special case are connection coefficients between two sets of orthonormal polynomials, another example…
In recent developments, a general approach for solving Riemann--Hilbert problems numerically has been developed. We review this numerical framework, and apply it to the calculation of orthogonal polynomials on the real line. Combining this…
We study two families of orthogonal polynomials. The first is a finite family related to the Askey-Wilson polynomials but the orthogonality is on the real line. A limiting case of this family is an infinite system of orthogonal polynomials…
We describe a refined version of the discrete Painlev\'e identification problem that emphasizes the importance on going beyond just the surface type in describing a discrete Painlev\'e dynamic. We give an example of solving such…
By using Fourier transforms of two symmetric sequences of finite orthogonal polynomials, we introduce two new classes of finite orthogonal functions and obtain their orthogonality relations via Parseval's identity.
In this paper we investigate polynomial interpolation using orthogonal polynomials. We use weight functions associated with orthogonal polynomials to define a weighted form of Lagrange interpolation. We introduce an upper bound of error…
It is well known that the zeros of orthogonal polynomials interlace. In this paper we study the case of multiple orthogonal polynomials. We recall known results and some recursion relations for multiple orthogonal polynomials. Our main…
Fourier transform of multivariate orthogonal polynomials on the unit ball are obtained. By using Parseval's identity, a new family of multivariate orthogonal functions are introduced. The results are expressed in terms of the continuous…
In this paper, we study the theory of orthogonal trigonometric polynomials (OTP). We obtain asymptotics of OTP with positive and analytic weight functions by Riemann-Hilbert approach and find they have relations with orthogonal polynomials…
For the orthogonal Lie algebra O(2n+1), in addition to the conventional set of orthogonal polynomials, another set is produced with the help of the Lie superalgebra OSP(1|2n). Difficulties related with expression of Dyson's constant for the…
Sets of orthogonal martingales are importants because they can be used as stochastic integrators in a kind of chaotic representation property, see [20]. In this paper, we revisited the problem studied by W. Schoutens in [21], investigating…
A high order linear $q$-difference equation with polynomial coefficients having $q$-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation is related to the number of orthogonality conditions that these…
Orthogonal polynomials for a family of weight functions on $[-1,1]^2$, $$ \CW_{\a,\b,\g}(x,y) = |x+y|^{2\a+1} |x-y|^{2\b+1} (1-x^2)^\g(1-y^2)^{\g}, $$ are studied and shown to be related to the Koornwinder polynomials defined on the region…