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Let $L$ be the $n$-th order linear differential operator $Ly = \phi_0y^{(n)}+ \phi_1y^{(n-1)} + \cdots + \phi_ny$ with variable coefficients. Arepresentation is given for $n$ linearly independent solutions of $Ly=\lambda ry$ as power series in $\lambda$, generalizing the SPPS (spectral parameterpower series) solution which has been previously developed for $n=2$. Thecoefficient functions in these series are obtained by recursively iterating asimple integration process, begining with a solution system for $\lambda=0$. Itis shown how to obtain such an initializing system working upwards fromequations of lower order. The values of the successive derivatives of the powerseries solutions at the basepoint of integration are given, which provides atechnique for numerical solution of $n$-th order initial value problems andspectral problems.

Spectral parameter power series for arbitrary order linear differential equations