Radial part calculations for quantum symmetric pairs with simple generators

We introduce the class of quantum symmetric pairs with simple generators. It is proved that the radial part of every element of a quantum symmetric pair with simple generators restricted to the set of regular points of this element can be computed. These computations are done explicitly for the Casimir elements of the quantum analogues of (SU(2), U(1)), (SU(2) x SU(2), diag) and (SU(3), U(2)) and give rise to second order q-difference equations for matrix valued spherical functions in general.