AI 中文总结
本文研究与等差数列指标集相关的广义Wronskian-Hermite多项式，验证其三个子类非零根为单根的猜想，解释z=0处根重数，证明其可表示Noumi-Yamada系统的有理解，并将结果用于多分量Hirota方程怪波模式的大参数渐近分析。
AI 中文摘要
本文研究与等差数列指标集相关的广义Wronskian-Hermite（WH）多项式，验证了这类多项式的三个子类的所有非零根均为单根的猜想，这三个子类在自然意义下覆盖了相关参数范围的一半以上；还通过杨图解释了z=0处的根重数，证明部分广义WH多项式可给出Noumi-Yamada系统（一类高阶Painlevé方程）有理解的表示；最后将所得结果应用于多分量Hirota方程的怪波模式大参数渐近分析。
英文摘要
In this paper, we study generalized Wronskian-Hermite (WH) polynomials associated with arithmetic-progression index sets. We verify the conjecture that all nonzero roots are simple for three subclasses of these polynomials, which, in a natural sense, cover more than half of the relevant parameter range. We also provide an interpretation of the root multiplicity at $z=0$ in terms of Young diagrams. In addition, we show that certain members of generalized WH polynomials provide representations of rational solutions of the Noumi-Yamada systems, a family of higher-order Painlevé equations. Finally, we apply our results to the large-parameter asymptotic analysis of rogue wave patterns for the multi-component Hirota equation.
