发表机构
School of Mathematics and Statistics, Nanfang College（南方学院数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对雅可比多项式，证明了含次数与参数的一致上界估计，验证了更强的Krasikov猜想并蕴含Erdélyi--Magnus--Nevai猜想，同时推导了相关函数与求积权的下界。
AI 中文摘要
设$p_n^{(α,β)}$为$[-1,1]$上关于权$(1-x)^α(1+x)^β$正交归一的雅可比多项式，其中$α,β\ge-1/2$，令$S=α+β+1$。我们证明了一致的次数-参数估计：$$(1-x)^{α+1/2}(1+x)^{β+1/2}\bigl|p_n^{(α,β)}(x)\bigr|^2\le C\max\bigl\{1,S^{1/3},S^{1/2}(n+1)^{-1/6}\bigr\}.$$该结果以等价的对称参数化形式证明了Krasikov提出的更强的次数敏感猜想，并蕴含Erdélyi--Magnus--Nevai猜想。证明从高参数象限的Krasikov估计出发，通过加权邻接关系将其传递到硬边缘，其奇异端点项相互抵消；直接超几何估计控制了剩余的端点帽区域。贝塞尔转向点论证表明，平方估计中的中间因子$S^{1/3}$不可省略。我们还推导了雅可比Christoffel函数与Gauss--Jacobi求积权的次数敏感下界。
英文摘要
Let $p_n^{(α,β)}$ denote the Jacobi polynomial orthonormal for the weight $(1-x)^α(1+x)^β$ on $[-1,1]$, where $α,β\ge-1/2$, and put $S=α+β+1$. We prove the uniform degree--parameter estimate $$ (1-x)^{α+1/2}(1+x)^{β+1/2} \bigl|p_n^{(α,β)}(x)\bigr|^2 \le C\max\bigl\{1,S^{1/3},S^{1/2}(n+1)^{-1/6}\bigr\}. $$ This proves, in an equivalent symmetric parametrisation, the stronger degree-sensitive conjecture proposed by Krasikov and implies the Erdélyi--Magnus--Nevai conjecture. The proof starts from Krasikov's estimate in the high-parameter quadrant and transports it to the hard edges through weighted contiguous relations whose singular endpoint terms cancel; direct hypergeometric estimates control the remaining endpoint caps. A Bessel turning-point argument shows that the intermediate factor $S^{1/3}$ in the squared estimate cannot be omitted. We also derive degree-sensitive lower bounds for Jacobi Christoffel functions and Gauss--Jacobi quadrature weights.
Comments18 pages
