  The principal character of a representation of the free group of rank two
into PSL(2, C) is a triple of complex numbers that determines an irreducible
representation uniquely up to conjugacy. It is a central problem in the
geometry of discrete groups and low dimensional topology to determine when such
a triple represents a discrete group that is not virtually abelian, that is a
Kleinian group. A classical necessary condition is J{\o}rgensen's inequality.
Here we use certainly shifted Chebyshev polynomials and trace identities to
determine new families of such inequalities, some of which are best possible.
The use of these polynomials also shows how we can identify the principal
character of some important subgroups from that of the group itself.
