论文标题
在带有应用的线性微分方程的多项式溶液上
On Polynomial Solutions of Linear Differential Equations with Applications
论文作者
论文摘要
对许多物理现象的分析可以简化为具有多项式系数的微分方程溶液的研究。在目前的工作中，我们为线性微分方程\ begin \ sum_ {k = 0}^{n}α_{k} \，r^，r^{k} {k} {k} \，y'（y''（y'（y''（y''） + \ sum_ {k = 0} {k = 0， r^{k} \，y'（r） - \ sum_ {k = 0}^{n -2}τ_{k} \，r^{k} \，y（r）= 0 \，，\ end，\ end \ end {equination*} for nutary $ n \ geq 2 $。我们以例如$ n \ ge 3 $的形式证明，必要的条件不足以确保存在多项式解决方案。使用Scheffé的标准，我们表明，从这个微分方程中，可以通过两项复发公式解决$ n $ generic方程。这些通用方程的封闭式解决方案是根据广义超几何函数给出的。对于任意的$ n $微分方程，开发了三个基本定理和一种算法来明确构建多项式解决方案。在$ n = 4 $的情况下，该算法用于建立多项式解决方案。为了证明这种方法的简单性和适用性，它用于研究Heun和Dirac方程的解决方案。
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.
