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Relative invariants for homogeneous linear differential equations. (English) Zbl 0693.34005

The author considers equations of the form: \(y^{(m)}+\sum^{m}_{j=1}c_ j(z)y^{(m-j)}=0,\) \(m\geq 3\), with coefficients which are meromorphic functions in some region \(\Omega\) of the complex plane. For these equations he gives algebraically independent relative invariants with respect to the transformations \(y=g(z)\cdot v\), \(z=f(\zeta)\), where g and f are analytic functions in some regions of the complex plane. His method is based on some new identities similar to those characterizing equations of the form \[ a(z)y^{''2}+b(z)y''y'+c(z)y''y+d(z)y^{'2}+e(z)y'y+f(z)y^ 2=0 \] whose solutions are free of movable branch points. It is also proved that the given equation can be transformed into a linear homogeneous equation having a fundamental system of local solutions of the form: \((\phi (z))^{m-1-i}(\psi (z))^ i,\) \(i=0,1,...,m-1\) with coefficients defined recursively as the polynomial combinations of \(c_ 1(z)\), \(c_ 2(z)\) and their derivatives.
Reviewer: A.Hajnosz

MSC:

34A30 Linear ordinary differential equations and systems
34M99 Ordinary differential equations in the complex domain
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