周期系数高阶线性微分方程的次正规解

考虑周期系数高阶线性微分方程f~((n))+∑j=1 nP_(n-j)(e~z)+Q_(n-j)(e~(-z))]f~((n-j))=R_1(e~z)+R_2(e~(-z)),其中n≥2,P_j(z),Q_j(z)(j=0,1,2,…,n-1),R_1(z)和R_2(z)均是关于z的多项式,且Pj(z),Qj(z)(j=0,1,2,…,n-1)不全为常数.在条件degPjdegP0(j=1,2,…,n-1)下,获得方程的次正规解的表示.

SUBNORMAL SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS

The representations of subnormal solutions for the higher order linear differential equation $$ f^{(n)}+\sum\limits_{j=1}^{n}P_{n-j} (\mathrm{e}^z)+ Q_{n-j} (\mathrm{e}^{-z})] f^{(n-j)} = R_1(\mathrm{e}^z) + R_2 (\mathrm{e}^{-z}) $$ are obtained, where $n\geq 2$, $P_j (z), Q_j (z)(j=0,1,2,\cdots, n-1)$, $R_1(z)$ and $R_2(z)$ are polynomials in $z$ such that $P_j (z), Q_j (z)$ $(j=0,1,2,\cdots, n-1)$ are not all constants, and $\deg P_{j}\deg P_0$ ,$(j=1,2,\cdots, n-1)$.