Related papers: Generalised Freud's equation and level densities w…

We discuss the recurrence coefficients of orthogonal polynomials with respect to a generalised sextic Freud weight \[\omega(x;t,\lambda)=|x|^{2\lambda+1}\exp\left(-x^6+tx^2\right),\qquad x\in\mathbb{R},\] with parameters $\lambda>-1$ and…

We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a generalized Freud weight \[w(x;t)=|x|^{2\lambda+1}\exp\left(-x^4+tx^2\right),\qquad x\in\mathbb{R},\] with parameters $\lambda>-1$…

We consider properties of semi-classical orthogonal polynomials with respect to the generalised Airy weight \[\omega(x;t,\lambda)=x^{\lambda}\exp\left(-\tfrac13x^3+tx\right),\qquad x\in \mathbb{R}^+,\] with parameters $\lambda>-1$ and $t\in…

We consider the semi-classical generalized Freud weight function \[w_{\lambda}(x;t) = |x|^{2\lambda+1}\exp(-x^4 +tx^2),\qquad x\in\mathbb{R},\] with $ \lambda>-1$ and $t\in\mathbb{R}$ parameters. We analyze the asymptotic behavior of the…

We discuss polynomials orthogonal with respect to a semi-classical generalised higher order Freud weight \[\omega(x;t,\lambda)=|x|^{2\lambda+1}\exp\left(tx^2-x^{2m}\right),\qquad x\in\mathbb{R},\] with parameters $\lambda > -1$,…

Approximations for an unknown density $g$ in terms of a reference density $f_\nu$ and its associated orthonormal polynomials are discussed. The main application is the approximation of the density $f$ of a sum $S$ of lognormals which may…

New sequences of orthogonal polynomials with respect to the weight functions $e^{-x} \rho_\nu(x),\ e^{- 1/x} x^{-1} \rho_{\nu} (x), \rho_{\nu}(x)= 2 x^{\nu/2} K_\nu(2\sqrt x),\ x >0, \nu \in \mathbb{R}$, where $K_\nu(z)$ is the modified…

Orthogonal polynomials for the weight $x^{\nu} \exp(-x - t/x),\ x, t > 0, \nu \in \mathbb{R}$ are investigated. Differential-difference equations, recurrence relations, explicit representations, generating functions and Rodrigues-type…

We consider random orthonormal polynomials $$ P_{n}(x)=\sum_{i=0}^{n}\xi_{i}p_{i}(x), $$ where $\xi_{0}$, . . . , $\xi_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep_0)$-moments, and…

We prove one-level density results for L-functions attached to primitive forms of level q, averaged over square-free q, conditional on the Generalized Riemann Hypothesis (GRH). We treat the even and odd orthogonal families separately and…

The generalisation of continuous orthogonal polynomial ensembles from random matrix theory to the $q$-lattice setting is considered. We take up the task of initiating a systematic study of the corresponding moments of the density from two…

The Cholesky factorization of the moment matrix is considered for discrete orthogonal polynomials of hypergeometrical type. We derive the Laguerre-Freud equations when the first moments of the weights are given by the ${}_1F_2$, ${}_2F_2$…

We consider orthogonal polynomials p_n with respect to an exponential weight function w(x) = exp(-P(x)). The related equations for the recurrence coefficients have been explored by many people, starting essentially with Laguerre [49], in…

Formulas are derived for the average level density of deformed, or transition, Gaussian orthogonal random matrix ensembles. After some general considerations about Gaussian ensembles we derive formulas for the average level density for (i)…

We consider the problem of stable sampling of multivariate real polynomials of large degree in a general framework where the polynomials are defined on an affine real algebraic variety $M$, equipped with a weighted measure. In particular,…

We show that for multivariate Freud-type weights $W_\alpha(x)=\exp(-|x|^\alpha)$, $\alpha>1$, any convex function $f$ on $R^d$ satisfying $fW_\alpha\in L_p(R^d)$ if $1\le p<\infty$, or $\lim_{|x|\to\infty}f(x)W_\alpha(x)=0$ if $p=\infty$,…

Generalized Jacobi polynomials are orthogonal polynomials related to a weight function which is smooth and positive on the whole interval of orthogonality up to a finite number of points, where algebraic singularities occur. The influence…

Orthogonal polynomials of degree $n$ with respect to the weight function $W_\mu(x) = (1-\|x\|^2)^\mu$ on the unit ball in $\RR^d$ are known to satisfy the partial differential equation $$ [ \Delta - \la x, \nabla \ra^2 - (2 \mu +d) \la x,…

Using the ratios theorems, we calculate the leading order terms in $N$ for the following averages of the characteristic polynomial and its derivative: $\left< \left|\Lambda_A(1 )\right| ^{r} \frac{ \Lambda_A'(\mathrm{e}^{\mathrm{i} \phi})…

We study the asymptotic properties of monic orthogonal polynomials (OPs) with respect to some Freud weights when the degree of the polynomial tends to infinity, including the asymptotics of the recurrence coefficients, the nontrivial…