# Please can someone explain the notation of this multivariate Taylor expansion?

Kamanzi-wa-Binyavanga, 2009, wrote the following paper, Calculating Cumulants of a Taylor Expansion of a Multivariate Function:

What I am confused about, is how precise the notation. I understand that we take each partial derivative with respect to some point ($$r,s,t$$ here) between $$1$$ and $$p$$, which might be the number of regressors in a statistical setting. What I don't understand is:

1. Why we must sum over each $$r,s,t$$ individually?
2. Why he sums over $$z$$ and $$s$$, not $$r$$ and $$s$$ in the double summation
3. Why he actually says each of these summations is from 1 to $$p_i$$. Am I correct in saying, the third for example, should look like $$\frac{1}{3!}\sum_{r=1}^{p_i}\sum_{s=1}^{p_i}\sum_{t=1}^{p_i}$$.
4. Lastly, this might seem simple, but if I wanted the Taylor expansion for the whole function $$Y$$, would I simply add $$\sum_{i=1}^{q}$$ before the term $$c$$?

Any help/clarity would be gratefully received. I'm trying to learn to use Einstein notation in this setting, but I need to understand exactly this expansion first.

2. It looks like a typo, since $$z$$ doesn't appear anywhere within that term
3. That's correct. $$p_i$$ is just the number of arguments (independent variables) of the function $$f_i$$
4. The whole variable $$Y$$ is a vector in $$\mathbb{R}^{q}$$ and each of its elements is explicitly defined as a function of certain $$p_i$$ variables (notation in the paper allows for them even to constitute nonoverlapping sets). If I understand correctly, in this setting you would compute expansion separately for each element (function). Aggregating them further doesn't make sense, in general