论文标题
四分之一平面的多结功能
Polyharmonic Functions in the Quarter Plane
论文作者
论文摘要
尽管离散的谐波函数已成为感兴趣的对象，但离散多谐波函数并非如此，例如在路径计数问题的渐近学中出现。在本文中，提出了一种用于计算四分之一平面中所有离散多谐波功能的新方法，以针对具有小步骤和零漂移的非单明模型。如果是有限组，则给出了使用解耦函数的替代方法，这通常会导致由有理函数组成的基础。以类似的方式，人们可以在连续环境中获得多谐功能，并且证明离散和连续情况之间的收敛性。最后，使用具体的示例，显示了为什么在无限组案例中似乎不起作用的解耦方法似乎不起作用。
While discrete harmonic functions have been objects of interest for quite some time, this is not the case for discrete polyharmonic functions, as appear for instance in the asymptotics of path counting problems. In this article, a novel method to compute all discrete polyharmonic functions in the quarter plane for non-singular models with small steps and zero drift is proposed. In case of a finite group, an alternative method using decoupling functions is given, which often leads to a basis consisting of rational functions. In a similar manner one can obtain polyharmonic functions in the continuous setting, and convergence between the discrete and continuous cases is proven. Lastly, using a concrete example it is shown why the decoupling approach seems not to work in the infinite group case.
