arXiv 2003.083242020-03-18math.CA

线性微分方程的多项式解及其应用

On Polynomial Solutions of Linear Differential Equations with Applications

Kyle R. Bryenton1, Andrew R. Cameron, Keegan L. A. Kirk, Nasser Saad, Patrick Strongman, Nikita Volodin

备注：: 18 pages

摘要：

对许多物理现象的分析可以归结为对多项式系数的微分方程的解的研究。本文建立了线性微分方程\begin{equation*} \sum_{k=0}^{n} α_{k} \, r^{k} \, y''(r) + \sum_{k=0}^{n-1} β_{k} \, r^{k} \, y'(r) - \sum_{k=0}^{n-2} τ_{k} \, r^{k} \, y(r) = 0 \, , \end{equation*}对任意$n\geq 2$存在多项式解的充要条件。通过实例证明了对于$n \ge 3$，所给出的必要条件不足以保证多项式解的存在性.使用Scheffé的标准，我们表明，从这个微分方程有$n$ -一般方程可解的两项递归公式。给出了广义超几何函数形式的一般方程的封闭解。对于任意$n$微分方程，给出了三个初等定理和一个算法，以显式构造多项式解。该算法用于建立$n=4$情况下的多项式解。为了证明这种方法的简单性和适用性，它被用来研究Heun和Dirac方程的解决方案。

英文摘要：

The analysis of many physical phenomena can be reduced to the study of solutions of differential equations with polynomial coefficients. In the present work, we establish the necessary and sufficient conditions for the existence of polynomial solutions to the linear differential equation \begin{equation*} \sum_{k=0}^{n} α_{k} \, r^{k} \, y''(r) + \sum_{k=0}^{n-1} β_{k} \, r^{k} \, y'(r) - \sum_{k=0}^{n-2} τ_{k} \, r^{k} \, y(r) = 0 \, , \end{equation*} for arbitrary $n\geq 2$. We show by example that for $n \ge 3$, the necessary condition is not enough to ensure the existence of the polynomial solutions. Using Scheffé's criteria, we show that from this differential equation there are $n$-generic equations solvable by a two-term recurrence formula. The closed-form solutions of these generic equations are given in terms of the generalized hypergeometric functions. For the arbitrary $n$ differential equations, three elementary theorems and one algorithm were developed to construct the polynomial solutions explicitly. The algorithm is used to establish the polynomial solutions in the case of $n=4$. To demonstrate the simplicity and applicability of this approach, it is used to study the solutions of Heun and Dirac equations.

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