  This work produces a q-analogue of the Cauchi-Szeg\"o integral representation
that retrieves a holomorphic function in the matrix ball from its values on the
Shilov boundary. Besides that, the Shilov boundary of the quantum matrix ball
is described and the U_q su(m,n)-covariance of the U_q s(u(m)x u(n))-invariant
integral on this boundary is established. The latter result allows one to
obtain a q-analogue for the principal degenerate series of unitary
representations related to the Shilov boundary of the matrix ball.
