Solving higher order Fuchs type differential systems avoiding the increase of the problem dimension (Q1321854)

Consider an ordinary differential equation \(t^ n x^{(n)}+ t^{n-1} P_{n-1}(t) x^{(n-1)}+\cdots+ P_ 0(t)x= 0\), where \(x\) is the \(m\)- vector unknown, and the coefficients \(P_ j(t)\) are \(m\times m\)-matrix valued analytic functions around \(t= 0\) with the expansion \(P_ j= \sum^ \infty_{k=0} P_{j,k} t^ k\). This paper is devoted to find solutions in the form \(X(t)= \left(\sum^ \infty_{ j=1} C_ j t^ j\right) t^ Z\), where \(Z\) is a \(p\times p\)-matrix, and \(C_ j\) are \(m\times p\)-matrices for some \(p\leq m\). By using a recurrence formula, the coefficients \(C_ j\) are explicitly given in terms of \(P_{j,k}\).