Related papers: Quadratic differentials and asymptotics of Laguerr…
We study the asymptotic behavior of Laguerre polynomials $L_n^{(\alpha_n)}(nz)$ as $n \to \infty$, where $\alpha_n$ is a sequence of negative parameters such that $-\alpha_n/n$ tends to a limit $A > 1$ as $n \to \infty$. These polynomials…
Laguerre and Laguerre-type polynomials are orthogonal polynomials on the interval $[0,\infty)$ with respect to a weight function of the form $w(x) = x^{\alpha} e^{-Q(x)}, Q(x) = \sum_{k=0}^m q_k x^k, \alpha > -1, q_m > 0$. The classical…
The classical orthogonal polynomials (Hermite, Laguerre and Jacobi) are involved in a vast number of applications in physics and engineering. When large degrees $n$ are needed, the use of recursion to compute the polynomials is not a good…
We consider polynomials orthogonal on $[0,\infty)$ with respect to Laguerre-type weights $w(x)=x^\alpha e^{-Q(x)}$, where $\alpha>-1$ and where $Q$ denotes a polynomial with positive leading coefficient. The main purpose of this paper is to…
Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials $L_{n}^{(\alpha)}(x)$, as well as complementary confluent hypergeometric functions. The expansions are valid for $n$ large and…
We consider Laguerre polynomials $L_n^{(\alpha_n)}(nz)$ with varying negative parameters $\alpha_n$, such that the limit $A = -\lim_n \alpha_n/n$ exists and belongs to $(0,1)$. For $A > 1$, it is known that the zeros accumulate along an…
We carry out the asymptotic analysis as $n \to \infty$ of a class of orthogonal polynomials $p_{n}(z)$ of degree $n$, defined with respect to the planar measure \begin{equation*} d\mu(z) = (1-|z|^{2})^{\alpha-1}|z-x|^{\gamma}\mathbf{1}_{|z|…
In this paper we consider the strong asymptotic behavior of Laguerre polynomials in the complex plane. The leading behavior is well known from Perron and Mehler-Heine formulas, but higher order coefficients, which are important in the…
In this paper we consider sequences of polynomials orthogonal with respect to certain discrete Laguerre-Sobolev inner product, with two perturbations (involving derivatives) located inside the oscillatory region for the classical Laguerre…
In this paper, We study the asymptotics of the leading coefficients and the recurrence coefficients for the orthogonal polynomials with repect to the Laguerre weight with singularity of root type and jump type at the soft edge via the…
Convergent expansions are derived for three types of orthogonal polynomials: Charlier, Laguerre and Jacobi. The expansions have asymptotic properties for large values of the degree. The expansions are given in terms of functions that are…
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…
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 study the asymptotic zero distribution of the rescaled Laguerre polynomials, $\displaystyle L_n^{(\alpha_n)}(nz)$, with the parameter $\alpha_n$ varying in such a way that $\displaystyle \lim_{n\rightarrow \infty}\alpha_n/n=-1$. The…
This paper deals with monic orthogonal polynomials generated by a Geronimus canonical spectral transformation of the Laguerre classical measure: \[ \frac{1}{x-c}x^{\alpha }e^{-x}dx+N\delta (x-c), \] for $x\in[0,\infty)$, $\alpha>-1$, a free…
We study polynomials that are orthogonal with respect to the modified Laguerre weight $z^{-n + \nu} e^{-Nz} (z-1)^{2b}$ in the limit where $n, N \to \infty$ with $N/n \to 1$ and $\nu$ is a fixed number in $\mathbb{R} \setminus…
In this work we study the Plancherel-Rotach type asymptotics for $q$-Laguerre orthogonal polynomials with complex scaling . The main term of the asymptotics contains Ramanujan function $A_{q}(z)$ for the scaling parameter on the vertical…
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 $s$…
In this paper, we consider the discrete Laguerre polynomials $P_{n, N}(z)$ orthogonal with respect to the weight function $w(x) = x^{\alpha} e^{-N cx}$ supported on the infinite nodes $L_N = \{ x_{k,N} = \frac{k^2}{N^2}, k \in \mathbb{N}…
We conclude our work [arXiv:2403.07628, arXiv:2503.12644] on asymptotic expansions at the soft edge for the classical $n$-dimensional Gaussian and Laguerre ensembles, now studying the gap-probability generating functions. We show that the…