Let $x$ and $y$ be vectors in $\mathbb{R}^2$. Define the function $g$ to be $$ g(x) = \|y-x\|^2. $$ This function can be expanded out as $$ g(x) = \|y\|^2 -2 \langle y,x \rangle + \|x\|^2. $$ I would like to write this in the binomial expansion type format: $$ g(x) = \sum_{j=0}^2 \begin{pmatrix}2\\j\end{pmatrix}(-1)^j||x||^j||y||^{2-j}. $$ However this summation is equal to $$ \sum_{j=0}^2 \begin{pmatrix}2\\j\end{pmatrix}(-1)^j\|x\|^j\|y\|^{2-j} = \|y\|^2 -2 \|y\| \ \|x\| + \|x\|^2, $$ which is not the same as the original expansion because the second term is a multiplication of norms instead of an inner product.
So is there some notation that accounts for this situation? How can I represent the original expansion of $g(x)$ in the form:
$$ g(x) = \sum_{j=0}^2 \begin{pmatrix}2\\j\end{pmatrix}(-1)^j??? $$
Similarly, for $\|y-x\|^4$ we have the expansion $$ |x+y|^4 = |y|^4 - 4 \langle x, y\rangle|y|^2 + (2|x|^2|y|^2 + 4\langle x, y \rangle^2) - 4 \langle x, y\rangle|x|^2 + |x|^4, $$ And again there is a visible correspondence with the regular binomial expansion. But does there exist notation that makes it possible to put it in a binomial type summation form?
As noted in the comment we cannot pretend to use the binomial formula in this case.
You have noted a ''similitude'' and we can see how this come from.
I suppose that we are using the $2-$norm, so we have: $$ ||x-y||=\sqrt{\sum_{i=1}^n(x_i-y_i)^2} $$
and: $$ ||x-y||^2=\sum_{i=1}^n(x_i-y_i)^2=\sum_{i=1}^n(x_i^2+y_i^2-2x_iy_i)=||x||^2+||y||^2-2 \sum_{i=1}^nx_iy_i= $$ and here it is clear because the binomial formula does not work, but we have:
$$||x-y||^2=||x||^2+||y||^2-2\langle x,y\rangle $$
Now we ca extend this result to any integer power $n$ as: $$ ||x-y||^n=\sqrt{\left[\sum_{i=1}^n(x_i-y_i)^2\right]^n}=\sqrt{\left[(||x||^2+||y||^2-2\langle x,y\rangle)\right]^n} $$