Let's look at the scalar product of two vectors $\mathbf{a}$ and $\mathbf{b}$:
$$ \mathbf{a} \cdot \mathbf{b} = \sum_ka_k \cdot b_k $$
I'm working my way through some physics problems where I have terms of the following form popping up:
$$ \sum_k a_k \cdot b_k \cdot c_k \cdot \ldots $$
This looks a little like a "generalization of the scalar product" to an arbitrary numbers of vectors to me. At the moment I'm making up my own notation by borrowing the "$\circ$" symbol from the Hadamard product to write
$$ \sum_ka_k \cdot b_k \cdot c_k \cdot \ldots = [ \mathbf{a} \circ \mathbf{b} \circ \mathbf{c} \circ \ldots ] $$
where I'm using square brackets to imply summation over all elements. Since I also have to deal with powers of the elements in this sum, I mimick Hadamard powers that I've seen written with the same symbol:
$$ \sum_k a^\alpha_k \cdot b^\beta_k \cdot c^\gamma_k \cdot \ldots = [ \mathbf{a}^{\circ\alpha} \circ \mathbf{b}^{\circ\beta} \circ \mathbf{c}^{\circ\gamma} \circ \ldots ] $$
However I would like to know if there is already an accepted way to represent a sum like the one above so I wouldn't have to reinvent the wheel. In addition, so far I have only seen the Hadamard product defined for matrices, not vectors, so it's my working assumption that this would be acceptable use for the symbol as well.
(I guess I could write this as a series of multiplications of diagonal matrices, however it feels a bit like overkill to introduce matrices if I know that none of my terms will ever have two indices to them. That's more personal taste though.)
There is a problem with your interpretation of these products as a generalization of the scalar product. I assume that $a_k$, $b_k$, $c_k...$ are the Cartesian components of vectors $\vec a$, $\vec b$, $\vec c...$ (i.e. in some orthonormal basis). Now, the scalar product of two vectors:
$$ S = \sum_k a_k\,b_k$$
is independent of the Cartesian system of coordinates used, i.e. $S$ is truly a scalar; it is a property only of the two vectors $\vec a$ and $\vec b$ used, and not of the coordinate system used. However, for three vectors $\vec a$, $\vec b$ and $\vec c$:
$$ Q = \sum_k a_k\,b_k\,c_k$$
is not a scalar. Its value depends on the Cartesian system of coordinates used. If $a'_k = \sum_jR_{kj}\,a_j$ are the Cartesian components of $\vec a$ in a rotated coordinate system ($R_{kj}$ are the elements of the rotation matrix), and similarly for $\vec b$ and $\vec c$, then in general:
$$ Q' =\sum_k a'_k\,b'_k\,c'_k \ne Q$$