  By using certain quantum differential operators, we construct a super
representation for the quantum queer supergroup U_v(q_n). The underlying space
of this representation is a deformed polynomial superalgebra in 2n^2 variables
whose homogeneous components can be used as the underlying spaces of queer
q-Schur superalgebras. We then extend the representation to its formal power
series algebra which contains a (super) submodule isomorphic to the regular
representation of U_v(q_n). A monomial basis M for U_v(q_n) plays a key role in
proving the isomorphism. In this way, we may present the quantum queer
supergroup U_v(q_n) by another new basis L together with some explicit
multiplication formulas by the generators. As an application, similar
presentations are obtained for queer q-Schur superalgebras via the above
mentioned homogeneous components.
  The existence of the bases M and L and the new presentation show that the
seminal construction of quantum gl_n established by
Beilinson-Lusztig-MacPherson thirty years ago extends to this "queer" quantum
supergroup via a completely different approach.
