  We define $q$-poly-Bernoulli polynomials $B_{n,\rho,q}^{(k)}(z)$ with a
parameter $\rho$, $q$-poly-Cauchy polynomials of the first kind
$c_{n,\rho,q}^{(k)}(z)$ and of the second kind $\widehat c_{n,\rho,q}^{(k)}(z)$
with a parameter $\rho$ by Jackson's integrals, which generalize the previously
known numbers and polynomials, including poly-Bernoulli numbers $B_n^{(k)}$ and
the poly-Cauchy numbers of the first kind $c_n^{(k)}$ and of the second kind
$\widehat c_n^{(k)}$. We investigate their properties connected with usual
Stirling numbers and weighted Stirling numbers. We also give the relations
between generalized poly-Bernoulli polynomials and two kinds of generalized
poly-Cauchy polynomials.
