I am trying to figure out what the attached picture's mathematical expressions mean. We have a kernel function $k$ that takes in 2 vectors that are $\in \mathbb{R}^d$.
I don't understand what the subscripts in the sum mean, could someone explain? What do the $j_1,...,j_d$ stand for? I also do not understand what the big parenthesis means. Could someone explain this to me as well?
It means $j_1, \ldots, j_d$ are nonnegative and sum to $p$.
and $$\binom{p}{j_1, \ldots, j_d} = \frac{p!}{j_1! j_2! \ldots j_d!}=\frac{p!}{\prod_{i=1}^d j_i!}$$
The multinomial coefficient.
Note that $$(u^Tv)^p = \left(\sum_{i=1}^d u^{(i)}v^{(i)} \right)^p$$