AI 中文总结
本研究针对支撑在半实轴上测度的有理修正的正交多项式，证明了其相对渐近式在复平面去掉半实轴的紧子集上的一致收敛性，明确了该渐近式由修正测度的有理函数的零点和极点及其重数决定
AI 中文摘要
给定关于支撑在正实轴$\boldsymbol{\nu}$上的正Borel测度$\nu$正交的正交多项式序列$\boldsymbol{\nu}$，设$\boldsymbol{\nu}$是关于修正测度$r(x)d\nu(x)$正交的正交多项式序列，其中$r$是某个有理函数，且$L_n(-1) = Q_n(-1)= (-1)^n$。本工作致力于证明相对渐近式$$ \frac{Q_n^{(d)}(z)}{L_n^{(d)}(z)} \text{当}n\to\boldsymbol{\nu} \text{时在}\boldsymbol{\nu}\text{的紧子集上一致收敛于}\boldsymbol{\nu}\text{，其中}a_k\text{和}b_j\text{是}r\text{的零点和极点，}A_k、B_j\text{分别是它们的重数。
英文摘要
Given a sequence of orthogonal polynomials $\{L_n\}_{n=0}^\infty$, orthogonal with respect to a positive Borel $ν$ measure supported on $\mathbb{R}_+$, let $\{Q_n\}_{n=0}^\infty$ be the sequence of orthogonal polynomials with respect to the modified measure $r(x)dν(x)$, where $r$ is certain rational function, and {$L_n(-1) = Q_n(-1)= (-1)^n$}. This work is devoted to the proof of the relative asymptotic $$ \frac{Q_n^{(d)}(z)}{L_n^{(d)}(z)} \unifn \prod_{k=1}^{N_1}\left(\frac{\sqrt{a_k}+i}{\sqrt{z}+\sqrt{a_k}}\right)^{A_k}\prod_{j=1}^{N_2} \left(\frac{\sqrt{z}+\sqrt{b_j}}{\sqrt{b_j}+i}\right)^{B_j},$$ on compact subsets of $\mathbb{C}\setminus\mathbb{R}_+$, where $a_k$ and $b_j$ are the zeros and poles of $r$, and the $A_k$, $B_j$ are their respective multiplicities.
Journal refMathematics (2024), 12(7), 1082
