Askey凸性猜想的一个证明

中文摘要 AI 辅助

对于$-1<\alpha\leq1/2$，设$J_\alpha$为第一类贝塞尔函数，$j_{\alpha,2}$为其第二个正零点。定义$\beta(\alpha)<\alpha+1$满足$$ \int_0^{j_{\alpha,2}}u^{-\beta(\alpha)}J_\alpha(u)\\,du=0. $$我们证明当$-1<\alpha\leq1/2$时$\beta''(\alpha)>0$，包括在$\alpha=1/2$处的左二阶导数。连续延拓$\beta(-1)=0$在$[-1,1/2]$上严格凸，加强了Askey 1993年的凸性猜想。解析论证使用了一个原函数的正展开式和一个消失的加权和。连续权重的递增比值给出负协方差项，而一个比较控制斜率和曲率。剩余的代数步骤通过有限精确多项式计算证明了八个有理不等式，这些计算在附录中重现。

英文摘要

For $-1<α\leq1/2$, let $J_α$ be the Bessel function of the first kind and let $j_{α,2}$ be its second positive zero. Define $β(α)<α+1$ by $$ \int_0^{j_{α,2}}u^{-β(α)}J_α(u)\,du=0. $$ We prove that $β''(α)>0$ for $-1<α\leq1/2$, including the left second derivative at $α=1/2$. The continuous extension $β(-1)=0$ is strictly convex on $[-1,1/2]$, strengthening Askey's 1993 convexity conjecture. The analytic argument uses a positive expansion of a primitive and a vanishing weighted sum. Increasing ratios of consecutive weights give a negative covariance term, while one comparison controls slope and curvature. The remaining algebraic step proves eight rational inequalities by finite exact polynomial calculations, reproduced in an appendix.

发表机构

• University of Coimbra（科英布拉大学）
• University of Porto（波尔图大学）

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