Related papers: Painlev\'e V and time dependent Jacobi polynomials

We show that the discrete Painlev\'e-type equations arising from quantum minimal surfaces are equations for recurrence coefficients of orthogonal polynomials for indefinite hermitian products. As a consequence, we obtain an explicit formula…

The discrete-time Toda equation arises as a universal equation for the relevant Hankel determinants associated with one-variable orthogonal polynomials through the mechanism of adjacency, which amounts to the inclusion of shifted weight…

We find and discuss asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with recurrence coefficients $a_{n}, b_{n}$. Our main goal is to consider the case where off-diagonal elements $a_{n}\to\infty$ as $n\to\infty$. Formulas…

We study reductions of the Korteweg--de Vries equation corresponding to stationary equations for symmetries from the noncommutative subalgebra. An equivalent system of $n$ second-order equations is obtained, which reduces to the Painlev\'e…

Our goal is to find an asymptotic behavior as $n\to\infty$ of orthogonal polynomials $P_{n}(z)$ defined by the Jacobi recurrence coefficients $a_{n}, b_{n}$. We suppose that the off-diagonal coefficients $a_{n}$ grow so rapidly that the…

A strategy is developed for writing the time-dependent Schr\"{o}dinger Equation (TDSE), and more generally the Dyson Series, as a convolution equation using recursive Fourier transforms, thereby decoupling the second-order integral from the…

We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on $[-1,1]$. The recurrence coefficients…

Orthogonal polynomials with respect to the weight function $w_{\beta,\gamma}(t) = t^\beta (1-t)^\gamma$, $\gamma > -1$, on the conic surface $\{(x,t): \|x\| = t, \, x \in \mathbb{R}^d, \, t \le 1\}$ are studied recently, and are shown to be…

A two-variable generalization of the Big $-1$ Jacobi polynomials is introduced and characterized. These bivariate polynomials are constructed as a coupled product of two univariate Big $-1$ Jacobi polynomials. Their orthogonality measure is…

The Littlewood-Paley theory is extended to weighted spaces of distributions on $[-1,1]$ with Jacobi weights $ \w(t)=(1-t)^\alpha(1+t)^\beta. $ Almost exponentially localized polynomial elements (needlets) $\{\phi_\xi\}$, $\{\psi_\xi\}$ are…

Persymmetric Jacobi matrices are invariant under reflection with respect to the anti-diagonal. The associated orthogonal polynomials have distinctive properties that are discussed. They are found in particular to be also orthogonal on the…

We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated to a certain normal matrix model. The model depends on a parameter and the asymptotic distribution of the eigenvalues undergoes a…

Applying the Jacobi method of second variation to the Bianchi IX system in Misner variables $(\alpha, \beta_+, \beta_-)$, we specialize to the Taub space background $(\beta_- = 0)$ and obtain the governing equations for linearized…

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 $\lambda > -1$,…

We revisit basics of classical Sturm-Liouville theory and, as an application, recover Bochner's classification of second order ODEs with polynomial coefficients and polynomial solutions by a new argument. We also outline how a wider class…

We look for differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with point masses…

We consider the Hankel determinant and orthogonal polynomials with respect to the deformed Laguerre weight $w(x; t) = {x^\alpha }{\mathrm e^{ - x}}{(x + t)^\lambda },\; x\in \mathbb{R}^{+} $ with parameters $\alpha > -1,\; t > 0$ and…

We study orthogonal polynomials with periodically modulated recurrence coefficients when $0$ lies on the hard edge of the spectrum of the corresponding periodic Jacobi matrix. In particular, we show that their orthogonality measure is…

We introduce a family of weight matrices $W$ of the form $T(t)T^*(t)$, $T(t)=e^{\mathscr{A}t}e^{\mathscr{D}t^2}$, where $\mathscr{A}$ is certain nilpotent matrix and $\mathscr{D}$ is a diagonal matrix with negative real entries. The weight…

We show that the Painleve equations P3-P5 can be derived (in a unified way) from a periodic sequence of Darboux transformations for a Schrodinger problem with quadratic eigenvalue dependency. The general problem naturally divides into three…