Three $q$-Painlev\'e type equations are derived through degenerations of the $q$-Painlev\'e equation with affine Weyl group symmetry of type $q$-$E_8^{ (1)}$. The three $q$-Painlev\'e type equations are associated with different realizations of the same symmetry/surface type $q$-$E_7^{ (1)}$/$q$-$A_1^{ (1)}$. We give three representations of affine Weyl group actions of type $q$-$E_7^{ (1)}$, and relations among them. The three $q$-$E_7^{ (1)}$ equations are also derived from the three representations respectively.
Categories: nlin.SI math-ph math.MP