Related papers: Monomial and Rodrigues orthogonal polynomials on t…
Skew orthogonal polynomials arise in the calculation of the $n$-point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry. In particular, the distribution functions are completely…
We construct a simple closed-form representation of degree-ordered system of bivariate Chebyshev-I orthogonal polynomials $\mathscr{T}_{n,r}(u,v,w)$ on simplicial domains. We show that these polynomials $\mathscr{T}_{n,r}(u,v,w),$…
We study a sequence of polynomials orthogonal with respect to a one parameter family of weights $$ w(x):=w(x,t)=\rex^{-t/x}\:x^{\al}(1-x)^{\bt},\quad t\geq 0, $$ defined for $x\in[0,1].$ If $t=0,$ this reduces to a shifted Jacobi weight.…
In this paper we study the following family of hypergeometric polynomials: $y_n(x) = \frac{ (-1)^\rho }{ n! } x^n {}_2 F_0(-n,\rho;-;-\frac{1}{x})$, depending on a parameter $\rho\in\mathbb{N}$. Differential equations of orders $\rho+1$ and…
We give a remarkable additional orthogonality property of the classical Legendre polynomials on the real interval $[-1,1]$: polynomials up to degree $n$ from this family are mutually orthogonal under the arcsine measure weighted by the…
Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line…
Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas, and two lines. For an integral with respect to an appropriate weight function defined on any…
Multiple Hermite polynomials are an extension of the classical Hermite polynomials for which orthogonality conditions are imposed with respect to $r>1$ normal (Gaussian) weights $w_j(x)=e^{-x^2+c_jx}$ with different means $c_j/2$, $1 \leq j…
In an isomorphic copy of the ring of symmetric polynomials we study some families of polynomials which are indexed by rational weight vectors. These families include well known symmetric polynomials, such as the elementary, homogeneous, and…
An explicit family of polynomials on the unit ball $B^d$ of $\RR^d$ is constructed, so that it is an orthonormal family with respect to the inner product $$ < f,g > = \rho \int_{B^d}\nabla f(x)\cdot \nabla g(x) dx + \CL (fg), $$ where $\rho…
In this paper we prove some characterizations of the matrix orthogonal polynomials whose derivatives are also orthogonal, which generalize other known ones in the scalar case. In particular, we prove that the corresponding orthogonality…
In this paper we give an asymptotic of the coefficients of the orthogonal polynomials on the unit circle, with respect of a weight of type $\displaystyle{ f : \theta \mapsto \prod_{1\le j \le M} \vert 1 - e^{i(\theta_{j}-\theta)}\vert…
Calogero-Sutherland models associated to the Weyl groups of type A and B with exchange terms included in the Hamiltonians systems have non-symmetric eigenfunctions, which are products of the ground state with members of a family of…
New sequences of discrete orthogonal polynomials associated with the modified Bessel function $K_\mu(z)$ or Macdonald function are considered. The corresponding weight function is $\lambda^k \rho_{k+\nu+1}(t)/ k!$, where $\ k \in…
In this work we classify all the order-two Hypergeometric operators $D$, symmetric with respect to some $2\times 2$ irreducible matrix-weight $W$ such that $DP_n=P_n\left(\begin{smallmatrix} \lambda_n&0\\0&\mu_n \end{smallmatrix} \right)$…
In this paper we introduce and discuss some classes of orthogonal polynomials in several non-commuting variables. The emphasis is on a non-commutative version of the orthogonal polynomials on the real line. We introduce recurrence equations…
We introduce orthogonal polynomials $M_j^{\mu,\ell}(x)$ as eigenfunctions of a certain self-adjoint fourth order differential operator depending on two parameters $\mu\in\mathbb{C}$ and $\ell\in\mathbb{N}_0$. These polynomials arise as…
There is a two-component log-gas system with Boltzmann factor which provides an interpolation between the eigenvalue PDF for $\beta = 1$ and $\beta = 4$ invariant random matrix ensembles. The solvability of this log-gas system relies on the…
We can write the polynomial solution of the second order linear differential equation of hypergeometric-type $$ \phi(x)y''+\psi(x)y'+\lambda y=0, $$ where $\phi$ and $\psi$ are polynomials, $\deg \phi\le 2$, $\deg \psi=1$ and $\lambda$ is a…
We investigate the symmetric Dunkl-classical orthogonal polynomials by using a new approach applied in connection with the Dunkl operator. The main aim of this technique is to determine the recurrence coefficients first and foremost. We…