Related papers: On the Asymptotics of Orthogonal Polynomials on Mu…

We investigate asymptotic behavior of polynomials $ Q_n(z) $ satisfying non-Hermitian orthogonality relations $$ \int_\Delta s^kQ_n(s)\rho(s)\dd s =0, \quad k\in\{0,\ldots,n-1\}, $$ where $ \Delta := [-a,a]\cup [-\ic b,\ic b] $, $ a,b>0 $,…

In this paper, we are interested in matrix valued orthogonal polynomials on the real line with respect to exponential weights. We obtain strong asymptotics as the degree tends to infinity in different regions of the complex plane, as well…

We investigate a Riemann-Hilbert problem (RHP), whose solution corresponds to a group of $q$-orthogonal polynomials studied earlier by Ismail et al. Using RHP theory we determine new asymptotic results in the limit as the degree of the…

We consider polynomials $P_n$ orthogonal with respect to the weight $J_{\nu}$ on $[0,\infty)$, where $J_{\nu}$ is the Bessel function of order $\nu$. Asheim and Huybrechs considered these polynomials in connection with complex Gaussian…

We obtain the strong asymptotics of polynomials $p_n(\lambda)$, $\lambda\in\mathbb{C}$, orthogonal with respect to measures in the complex plane of the form $$ e^{-N(|\lambda|^{2s}-t\lambda^s-\overline{t\lambda}^s)}dA(\lambda), $$ where $s$…

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…

In this paper, we present a new method via the transfer matrix approach to obtain asymptotic formulae of orthogonal polynomials with asymptotically identical coefficients of bounded variation. We make use of the hyperbolicity of the…

Mehler-Heine asymptotics describe the behavior of orthogonal polynomials near the edges of the interval where the orthogonality measure is supported. For Jacobi polynomials and Laguerre polynomials this asymptotic behavior near the hard…

We study orthogonal polynomials and Hankel determinants generated by a symmetric semi-classical Jacobi weight. By using the ladder operator technique, we derive the second-order nonlinear difference equations satisfied by the recurrence…

We develop a numerical method for computing with orthogonal polynomials that are orthogonal on multiple, disjoint intervals for which analytical formulae are currently unknown. Our approach exploits the Fokas--Its--Kitaev Riemann--Hilbert…

We consider the large-$N$ asymptotics of a system of discrete orthogonal polynomials on an infinite regular lattice of mesh $\frac{1}{N}$, with weight $e^{-NV(x)}$, where $V(x)$ is a real analytic function with sufficient growth at…

In this paper we compute the asymptotic behavior of the recurrence coefficients for polynomials orthogonal with respect to a logarithmic weight $w(x){\rm d}x = \log \frac{2k}{1-x}{\rm d}x$ on $(-1,1)$, $k > 1$, and verify a conjecture of…

We study the properties and asymptotics of the Jacobi matrices associated with equilibrium measures of the weakly equilibrium Cantor sets. These family of Cantor sets were defined and different aspects of orthogonal polynomials on them were…

Using the steepest descent method for oscillatory Riemann-Hilbert problems introduced by Deift and Zhou [Ann. Math. {\bf 137}(1993), 295-368], we derive asymptotic formulas for the Meixner polynomials in two regions of the complex plane…

We obtain Plancherel-Rotach type asymptotics valid in all regions of the complex plane for orthogonal polynomials with varying weights of the form $e^{-NV(x)}$ on the real line, assuming that $V$ has only two Lipschitz continuous…

In this paper we deal with polynomials orthogonal with respect to an inner product involving derivatives, that is, a Sobolev inner product. Indeed, we consider Sobolev type polynomials which are orthogonal with respect to $$(f,g)=\int fg…

We consider the orthogonal polynomials on $[-1,1]$ with respect to the weight $$ w_c(x)=h(x)(1-x)^{\alpha}(1+x)^{\beta} \Xi_{c}(x), \quad \alpha, \beta >-1, $$ where $h$ is real analytic and strictly positive on $[-1, 1]$, and $\Xi_{c}$ is…

We continue studying polynomials generated by the Szeg\H{o} recursion when a finite number of Verblunsky coefficients lie outside the closed unit disk. We prove some asymptotic results for the corresponding orthogonal polynomials and then…

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…

We consider asymptotics of planar orthogonal polynomials $P_{n,N}$ (where $\mathrm{deg}P_{n,N}=n$) with respect to the weight $$\frac{|z-w|^{2NQ_1}}{(1+|z|^2)^{N(1+Q_0+Q_1)+1}}, \quad(Q_0,Q_1 > 0)$$ in the whole complex plane. With $n,…