I want to prove that for any strict partition $\lambda$ of length $\ell \in [1, n]$, we have $$Q_{\lambda}(x_1, \cdots, x_n) = 2^{\ell}\sum_{w \in S_n}sgn(w)\cdot w\biggl(x^{\lambda_1}_1 \cdots x^{\lambda_\ell}_\ell \prod^{\ell}_{i = 1}\prod^{n}_{j = i + 1}(1 + x^{-1}_ix_j) \biggr), \hspace{4mm} (1)$$ where $Q_{\lambda}(x_1, \cdots, x_n)$ is a Schur $Q-$function.
All I have been able to prove is that for any strict partition $\lambda$ of length $\ell \leq n$, we have $$Q_{\lambda}(x_1, \cdots, x_n) = 2^{\ell}\sum_{w \in S_n/S_{n - \ell}}w\biggl(x^{\lambda_1}_1 \cdots x^{\lambda_\ell}_\ell \prod^{\ell}_{i = 1}\prod^{n}_{j = i + 1}\frac{x_i + x_j}{x_i - x_j} \biggr), \hspace{5mm} (2)$$ where $S_{n - \ell}$ is the subgroup of permutations in $S_n$ which fix $1, 2, \cdots, \ell$.
Any hints on how to get $(1)$ from $(2)$?
Thanks!