  We present an operator approach to deriving Mehler's formula and the Rogers
formula for the bivariate Rogers-Szeg\"{o} polynomials $h_n(x,y|q)$. The proof
of Mehler's formula can be considered as a new approach to the nonsymmetric
Poisson kernel formula for the continuous big $q$-Hermite polynomials
$H_n(x;a|q)$ due to Askey, Rahman and Suslov. Mehler's formula for $h_n(x,y|q)$
involves a ${}_3\phi_2$ sum and the Rogers formula involves a ${}_2\phi_1$ sum.
The proofs of these results are based on parameter augmentation with respect to
the $q$-exponential operator and the homogeneous $q$-shift operator in two
variables. By extending recent results on the Rogers-Szeg\"{o} polynomials
$h_n(x|q)$ due to Hou, Lascoux and Mu, we obtain another Rogers-type formula
for $h_n(x,y|q)$. Finally, we give a change of base formula for $H_n(x;a|q)$
which can be used to evaluate some integrals by using the Askey-Wilson
integral.
