A generalized Prouhet–Thue–Morse identity for products of binary sums

Let $k\geq 1$ and let $a_{1},\ldots,a_{k}$ be nonnegative integers. Put $A=(a_{1},\ldots,a_{k})$ and let $X=(x,x_{1},\ldots,x_{k})$ be a vector of variables. Define

$$G_{A}(X)=\sum_{i_{1}=0}^{2^{a_{1}}-1}\cdots\sum_{i_{k}=0}^{2^{a_{k}}-1}(-1)^{\sum_{j=1}^{k}s_{2}(i_{j})}\Bigl(x+\sum_{j=1}^{k}i_{j}x_{j}\Bigr)^{\sum_{j=1}^{k}a_{j}}.$$

The generalized Prouhet–Thue–Morse identity. One has

$$G_{A}(X)=(-1)^{\sum_{j=1}^{k}a_{j}}2^{\sum_{j=1}^{k}\frac{a_{j}(a_{j}-1)}{2}}\Bigl(\sum_{j=1}^{k}a_{j}\Bigr)!\prod_{j=1}^{k}x_{j}^{a_{j}}.$$

This is proposed as a generalization of identities proved earlier in the paper for the Prouhet–Thue–Morse sequence. Its status is not established in the supplied text.