  In the framework of (vector valued) quantized holomorphic functions defined
on non-commutative spaces, ``quantized hermitian symmetric spaces'', we analyze
what the algebras of quantized differential operators with variable
coefficients should be. It is an emediate point that even $0$th order
operators, given as multiplications by polynomials, have to be specified as
e.g. left or right multiplication operators since the polynomial algebras are
replaced by quadratic, non-commutative algebras. In the settings we are
interested in, there are bilinear pairings which allows us to define
differential operators as duals of multiplication operators. Indeed, there are
different choices of pairings which lead to quite different results. We
consider three different pairings. The pairings are between quantized
generalized Verma modules and quantized holomorphically induced modules. It is
a natural demand that the corresponding representations can be expressed by
(matrix valued) differential operators. We show that a quantum Weyl algebra
${\mathcal W}eyl_q(n,n)$ introduced by T. Hyashi (Comm. Math. Phys. 1990) plays
a fundamental role. In fact, for one pairing, the algebra of differential
operators, though inherently depending on a choice of basis, is precisely
matrices over ${\mathcal W}eyl_q(n,n)$. We determine explicitly the form of the
(quantum) holomorphically induced representations and determine, for the
different pairings, if they can be expressed by differential operators.
