On the Chebyshev property for a new family of functions

We analyze whether a given set of analytic functions is an Extended Chebyshev system. This family of functions appears studying the number of limit cycles bifurcating from some nonlinear vector field in the plane. Our approach is mainly based on the so called Derivation–Division algorithm. We prove that under some natural hypotheses our family is an Extended Chebyshev system and when some of them are not fulfilled then the set of functions is not necessarily an Extended Chebyshev system. One of these examples constitutes an Extended Chebyshev system with high accuracy.