Generalized Schwarzian derivatives and higher order differential equations

Abstract

It is shown that the well-known connection between the second order linear differential equation h + B(z) h = 0, with a solution base {h 1 , h 2 }, and the Schwarzian derivativeThis generalization depends upon an appropriate definition of the generalized Schwarzian derivative S k ( f ) of a function f which is induced by k −1 ratios of linearly independent solutions of h (k) + B(z) h = 0. The class R k ( ) of meromorphic functions f such that S k ( f ) is analytic in a given domain is also completely described. It is shown that if is the unit disc D or the complex plane C, then the order of growth of f ∈ R k ( ) is precisely determined by the growth of S k ( f ), and vice versa. Also the oscillation of solutions of h (k) + B(z) h = 0, with the analytic coefficient B in D or C, in terms of the exponent of convergence of solutions is briefly discussed.

Schwarzian conditions for linear differential operators with selected differential Galois groups

“…When dealing with linear differential operators, we have seen the emergence of Schwarzian derivatives, consequence of the fact that the Schwarzian derivative is appropriate for the com position of functions [19] (see the chain rule of the Schwarzian derivative of the composition of function). Do higher order Schwarzian derivatives [55][56][57]58] occur for pullback-symmetries of nonlinear ODE's, or, more generally, for functional equations?…”