Related papers: Generalized Airy polynomials, Hankel determinants …
Based on the work of Chen and Its [{\em J. Approx. Theory} {\bf 162} ({2010}) {270--297}], we further study orthogonal polynomials with respect to the singularly perturbed Laguerre weight $w(x;t,\alpha) = {x^\alpha}{\mathrm e^{-…
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 the monic orthogonal polynomials with respect to a singularly perturbed Airy weight. By using Chen and Ismail's ladder operator approach, we derive a discrete system satisfied by the recurrence coefficients for the orthogonal…
We study the Hankel determinant and orthogonal polynomials with respect to the two-parameter weight function $$ w(x)=w(x;t_1, t_2):=\exp(-x^6-t_2 x^4-t_1 x^2),\qquad x\in\mathbb{R}, $$ with $t_1,\; t_2 \in \mathbb{R}$. This problem arises…
We study the monic polynomials orthogonal with respect to a symmetric perturbed Gaussian weight $$ w(x;t):=\mathrm{e}^{-x^2}\left(1+t\: x^2\right)^\lambda,\qquad x\in \mathbb{R}, $$ where $t> 0,\;\lambda\in \mathbb{R}$. This weight is…
We are concerned with the monic orthogonal polynomials with respect to a singularly perturbed Laguerre-type weight. By using the ladder operator approach, we derive a complicated system of nonlinear second-order difference equations…
In this paper, we study the orthogonal polynomials with respect to a singularly perturbed Pollaczek-Jacobi type weight $$ w(x,t):=(1-x^2)^\alpha\mathrm{e}^{-\frac{t}{1-x^{2}}},\qquad x\in[-1,1],\;\;\alpha>0,\;\;t>0. $$ By using the ladder…
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…
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 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.…
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 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…
We compute asymptotics for Hankel determinants and orthogonal polynomials with respect to a discontinuous Gaussian weight, in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support.…
In this paper we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, A_n(z) and B_n(z), appearing in the ladder operators satisfy the two…
This paper studies the monic semi-classical Laguerre polynomials based on previous work by Boelen and Van Assche \cite{Boelen}, Filipuk et al. \cite{Filipuk} and Clarkson and Jordaan \cite{Clarkson}. Filipuk, Van Assche and Zhang proved…
We study the Hankel determinant generated by the moments of the deformed Laguerre weight function $x^{\alpha}{\rm{e}}^{-x}\prod\limits_{k=1}^{N}(x+t_k)^{\lambda_k}$, where $x\in \left[0,+\infty \right)$, $\alpha,t_k >0,…
We study the Hankel determinant generated by a Gaussian weight with Fisher-Hartwig singularities of root type at $t_j$, $j=1,\cdots ,N$. It characterizes a type of average characteristic polynomial of matrices from Gaussian unitary…
We study the Hankel determinant generated by the Gaussian weight with jump discontinuities at $t_1,\cdots,t_m$. By making use of a pair of ladder operators satisfied by the associated monic orthogonal polynomials and three supplementary…
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…
E. Heine in the 19th century studied a system of orthogonal polynomials associated with the weight $\left[x(x-\alpha)(x-\beta)\right]^{-\frac{1}{2}}$, $x\in[0,\alpha]$, $0<\alpha<\beta$. A related system was studied by C. J. Rees in 1945,…