What is this mathematical expression?

Question

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?

This may help you: https://en.wikipedia.org/wiki/Multinomial_theorem — Sam Streeter, Oct 31 '18 at 15:40
Thanks! I had a hard time searching because I did not know what the name of this expression is. — 24n8, Oct 31 '18 at 15:42

Siong Thye Goh · Accepted Answer · 2018-10-31T15:49:19.740

1

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 $$(u^Tv)^p = \left(\sum_{i=1}^d u^{(i)}v^{(i)} \right)^p$$

edited Oct 31 '18 at 15:49

answered Oct 31 '18 at 15:42

Siong Thye Goh

149,520
20
88
149

Why does the sum subscripts mean the j's are non-negative? – 24n8 Oct 31 '18 at 15:43
1

it didn't say that explicitly, but since I know the multinomial theorem... – Siong Thye Goh Oct 31 '18 at 15:44
Ohh. So does $j_1,\ldots,j_d$ take on the values of $1,\ldots,d$, respecitlvey? – 24n8 Oct 31 '18 at 15:45
1

they have to sum up to $p$, so nope. – Siong Thye Goh Oct 31 '18 at 15:46
for example if $d=3$, and $p=5$, some of the indices are $(5, 0,0), (1,2,2)$, $(3,0,2)$, and so on. – Siong Thye Goh Oct 31 '18 at 15:48
Ah okay. And is there a typo in your last edit? There shouldn't be a p on the RHS, or the LHS is missing a p. – 24n8 Oct 31 '18 at 15:48
you are right, thanks. – Siong Thye Goh Oct 31 '18 at 15:49
One more question. You gave the example for $d=3$ and $p=5$. So $j_1,\ldots,j_d$ the set of all indices that satisfy $j_1 + \ldots + j_d = p$? – 24n8 Oct 31 '18 at 15:53
yes, that is correct. – Siong Thye Goh Oct 31 '18 at 15:55
Is there a quick way to modify the multinomial expression for $(u^Tv)^p$ to $(u^Tv+1)^p$ – 24n8 Oct 31 '18 at 17:25
Let us continue this discussion in chat. – Siong Thye Goh Oct 31 '18 at 17:33

What is this mathematical expression?

1 Answers1