用无限阶矩阵求微分方程在奇点处的级数解

Series Solutions of Linear Ordinary Differential Equation at Singular Point by Infinite Order Matrix

应用线性微分算子在幂基下的无限阶矩阵,研究线性微分方程在奇点处的级数解.得到一个计算无限阶矩阵属于零的特征向量的递推公式,进而用这些特征向量表示级数解.给出用有限阶矩阵判断奇点正则性的方法,并改进了Fuchs定理.

The series solutions of the linear ordinary differential equation at singular point were studied via the infinite order matrix of the linear differential operator in power series basis. We got a recurrence formula to compute the characteristic vectors of the infinite order matrix belonging to λ = 0 and then completed the expression of the series solutions with the characteristic vectors. The regularity of singular point is judged with a finite order matrix, and the Fuchs theorem has been improved.