相关论文: Asymptotics of polynomials orthogonal over circula…

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|…

Let $w$ be a weight on the unit disk $\mathbb{D}$ having the form \[w(z)=|v(z)|^2\prod_{k=1}^s\left|\frac{z-a_k}{1-z\overline{a}_k}\right|^{m_k}\,,\quad m_k>-2,\ |a_k|<1,\] where $v$ is analytic and free of zeros in $\overline{\mathbb{D}}$,…

We study the asymptotic behavior of the Bergman orthogonal polynomials $(p_n)_{n=0}^{\infty}$ for a class of bounded simply connected domains $D$. The class is defined by the requirement that conformal maps $\varphi$ of $D$ onto the unit…

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 $s$…

Asymptotic expansions are derived for Gegenbauer (ultraspherical) polynomials for large order $n$ that are uniformly valid for unbounded complex values of the argument $z$, including the real interval $0 \leq z \leq 1$ in which the zeros in…

We consider the orthogonal polynomials $\{P_{n}(z)\}$ with respect to the measure $|z-a|^{2N c} {\rm e}^{-N |z|^2} \,{\rm d} A(z)$ over the whole complex plane. We obtain the strong asymptotic of the orthogonal polynomials in the complex…

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 obtain asymptotics of polynomials satisfying the orthogonality relations $$ \int_{\mathbb{R}} z^k P_n(z; t , N) \mathrm{e}^{-N \left(\frac{1}{4}z^4 + \frac{t}{2}z^2 \right)} \mathrm{d} z = 0 \quad \text{ for } \quad k = 0, 1, ..., n-1,…

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

This paper complements the recent investigation of \cite{DM} on the asymptotic behavior of polynomials orthogonal over the interior of an analytic Jordan curve $L$. We study the specific case of $L=\{z= w-1 +(w-1)^{-1},\ |w|=R\}$, for some…

We establish the asymptotic zero distribution for polynomials generated by a four-term recurrence relation with varying recurrence coefficients having a particular limiting behavior. The proof is based on ratio asymptotics for these…

We consider the orthogonal polynomials, $\{P_n(z)\}_{n=0,1,\cdots}$, with respect to the measure $$|z-a|^{2c} e^{-N|z|^2}dA(z)$$ supported over the whole complex plane, where $a>0$, $N>0$ and $c>-1$. We look at the scaling limit where $n$…

Asymptotic expansions are given for large values of $n$ of the generalized Bessel polynomials $Y_n^\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given…

We consider the Bernoulli polynomials of the second kind, which can be related to the generalised Bernoulli polynomials $B_n^{(n)}(z)$. The asymptotic expansions of the scaled polynomials $B_n^{(n)}(nz)$ are obtained as $n\to\infty$ when…

We consider the planar orthogonal polynomial $p_{n}(z)$ with respect to the measure supported on the whole complex plane $${\rm e}^{-N|z|^2} \prod_{j=1}^\nu |z-a_j|^{2c_j}\,{\rm d} A(z)$$ where ${\rm d} A$ is the Lebesgue measure of the…

We compute the pointwise asymptotics of orthogonal polynomials with respect to a general class of pure point measures supported on finite sets as both the number of nodes of the measure and also the degree of the orthogonal polynomials…

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,…

We study a family of monic orthogonal polynomials which are orthogonal with respect to the varying, complex valued weight function, $\exp(nsz)$, over the interval $[-1,1]$, where $s\in\mathbb{C}$ is arbitrary. This family of polynomials…

We study asymptotic behavior of orthogonal polynomials on the unit circle with varying Verblunsky coefficients $\alpha_{n,N}$ when the ratio $n/N$ converges as $n,N\to\infty$. First, we give a streamlined proof of ratio asymptotics for…

Let ${z_n}$ be a sequence in the unit disk ${z\in\mathbb{C}:|z|<1}$. It is known that there exists a unique positive Borel measure in the unit circle ${z\in\mathbb{C}:|z|=1}$ such that the orthogonal polynomials ${\Phi_n}$ satisfy…