Zero Distribution of Composite Polynomials and Polynomials Biorthogonal to Exponentials期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Zero Distribution of Composite Polynomials and Polynomials Biorthogonal to Exponentials

Authors:

Email author" target="_blank">D?S?Lubinsky Email author A?Sidi

Institution:

(1) School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA;(2) Department of Computer Science, Technion-Israel Institute of Technology, Haifa, 32000, Israel

Abstract:

We analyze polynomials P _n that are biorthogonal to exponentials $\{e^{-\sigma _{n,j}x}\}_{j=1}^{n}$ , in the sense that

$\int_{0}^{\infty }P_{n}(x)e^{-\sigma _{n,j}x}x^{\alpha }\,dx=0,\quad 1\leq j\leq n.$

Here α>−1. We show that the zero distribution of P _n as n→∞ is closely related to that of the associated exponent polynomial

$Q_{n}(y)=\prod\limits_{j=1}^{n}(y+1/\sigma _{n,j})=\sum_{j=0}^{n}q_{n,j}y^{j}.$

More precisely, we show that the zero counting measures of {P _n(−4nx)}_n=1^∞ converge weakly if and only if the zero counting measures of {Q _n}_n=1^∞ converge weakly. A key step is relating the zero distribution of such a polynomial to that of the composite polynomial

$\sum_{j=0}^{n}q_{n,j}\Delta _{n,j}x^{j},$

under appropriate assumptions on {Δ_n,j}.

Keywords:

Biorthogonal polynomials Zero distribution Laguerre polynomials

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