Related papers: Zeros of Quasi-Orthogonal Jacobi Polynomials

We investigate the large $n$ behavior of Jacobi polynomials with varying parameters $P_{n}^{(an+\alpha,\,bn+\beta)}(1-2\lambda^{2})$ for $a,b >-1$ and $\lambda\in(0,\,1)$. This is a well-studied topic in the literature but some of the…

Jacobi polynomials are polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two sl_2 irreducible modules. We study sequences of r polynomials whose zeros form the unique solution of the Bethe Ansatz…

A Bernstein type inequality is obtained for the Jacobi polynomials $P_n^{\alpha,\beta}(x)$, which is uniform for all degrees $n\ge0$, all real $\alpha,\beta\ge0$, and all values $x\in [-1,1]$. It provides uniform bounds on a complete set of…

We discuss some properties of the moduli of smoothness with Jacobi weights that we have recently introduced and that are defined as \[ \omega_{k,r}^\varphi(f^{(r)},t)_{\alpha,\beta,p} :=\sup_{0\leq h\leq t} \left\|…

For $0<\alpha<1$, we study the zeros of the sequence of polynomials $\left\{ P_{m}(z)\right\} _{m=0}^{\infty}$ generated by the reciprocal of $(1-t)^{\alpha}(1-2zt+t^{2})$, expanded as a power series in $t$. Equivalently, this sequence is…

We investigate asymptotic behavior of polynomials $p^{\omega}_n(z)$ satisfying varying non-Hermitian orthogonality relations $$ \int_{-1}^{1} x^kp^{\omega}_n(x)h(x) e^{\mathrm{i} \omega x}\mathrm{d} x =0, \quad k\in\{0,\ldots,n-1\}, $$…

The Jacobi polynomials $\hat{P}_n^{(\alpha,\beta)}(x)$ conform the canonical family of hypergeometric orthogonal polynomials (HOPs) with the two-parameter weight function $(1-x)^\alpha (1+x)^\beta, \alpha,\beta>-1,$ on the interval…

The purpose of this note is to extend in a simple and unified way the known results on interlacing of zeros of paraorthogonal polynomials on the unit circle. These polynomials can be regarded as the characteristic polynomials of any matrix…

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 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 $,…

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…

By applying the Euler--Rayleigh methods to a specific representation of the Jacobi polynomials as hypergeometric functions, we obtain new bounds for their largest zeros. In particular, we derive upper and lower bound for…

For a given $\theta\in (-1,1)$, we find out all parameters $\alpha,\beta\in \{0,1\}$ such that, there exists a linear combination of Jacobi polynomials $J_{n+1}^{(\alpha,\beta)}(x)-C J_{n}^{(\alpha,\beta)}(x)$ which generates a Lobatto…

We show that the zeros of consecutive orthogonal polynomials $p_n$ and $p_{n-1}$ are linearly connected by a doubly stochastic matrix for which the entries are explicitly computed in terms of Christoffel numbers. We give similar results for…

We show how to obtain linear combinations of polynomials in an orthogonal sequence $\{P_n\}_{n\geq 0}$, such as $Q_{n,k}(x)=\sum\limits_{i=0}^k a_{n,i}P_{n-i}(x)$, $a_{n,0}a_{n,k}\neq0$, that characterize quasi-orthogonal polynomials of…

The literature on quaternionic polynomials and, in particular, on methods for determining and classifying their zero-sets, is fast developing and reveals a growing interest on this subject. In contrast, polynomials defined over the algebra…

In this article, we study the quasi-homogeneity of a superpotential in a complete free algebra over an algebraic closed field of characteristic zero. We prove that a superpotential with finite dimensional Jacobi algebra is right equivalent…

The kernel polynomial method based on Jacobi polynomials $P_n^{\alpha,\beta}(x)$ is proposed. The optimal-resolution positivity-preserving kernels and the corresponding damping factors are obtained. The results provide a generalization of…

Sequences of orthogonal polynomials that are alternative to the Jacobi polynomials on the interval $[0,1]$ are defined and their properties are established. An $(\alpha,\beta)$-parameterized system of orthogonal polynomials of the…

The multi-indexed Jacobi polynomials are the main part of the eigenfunctions of exactly solvable quantum mechanical systems obtained by certain deformations of the P\"oschl-Teller potential (Odake-Sasaki). By fine-tuning the parameter(s) of…