The bounded Littlewood identities

Let $m,n$ be nonnegative integers. For the coefficients $\kappa^{(1)}_{\lambda}$ and $\kappa^{(2)}_{\lambda}$ defined for partitions with empty $2$-core, one has

$$\sum_{\lambda\in P_{+}(n)}\kappa^{(1)}_{\lambda}(q^{-m};q,t)P_{\lambda}(x;q,t)=(x_1\cdots x_n)^mK_{(m^n)}(x;q,t;q^{1/2},-q^{1/2},t^{1/2},-t^{1/2}),$$

and

$$\sum_{\lambda\in P_{+}(n)}\kappa^{(2)}_{\lambda}(q^{-m};q,t)P_{\lambda}(x;q,t)=(x_1\cdots x_n)^mK_{(m^n)}(x;q,t;1,-1,(qt)^{1/2},-(qt)^{1/2}).$$

Bounded Littlewood identities. Both displayed identities hold.