Clarification over product of products $\prod$ notation

Question

This might be a trivial question to ask in this forum but I would like some clarification over a particular formula. Suppose we are given $$f^{eq}_i=\rho w_i\prod_{j=1}^3\bigg(2-\sqrt{1+3u_j^2}\bigg)\bigg(\frac{2u_j+\sqrt{1+3u_j^2}}{1-u_j}\bigg)^{c_{i,j}}.$$ Then what I would like to know is that if we write $$p1=\bigg(2-\sqrt{1+3u_1^2}\bigg)\bigg(\frac{2u_1+\sqrt{1+3u_1^2}}{1-u_1}\bigg)^{c_{i,1}},$$ $$p2=\bigg(2-\sqrt{1+3u_2^2}\bigg)\bigg(\frac{2u_2+\sqrt{1+3u_2^2}}{1-u_2}\bigg)^{c_{i,2}}$$ and $$p3=\bigg(2-\sqrt{1+3u_3^2}\bigg)\bigg(\frac{2u_3+\sqrt{1+3u_3^2}}{1-u_3}\bigg)^{c_{i,3}},$$ can we then write $f^{eq}_i=\rho w_ip1p2p3$. My main confusion is with regards to the definition of $\prod$ notation. This formula is taken from : http://arxiv.org/pdf/cond-mat/0311156.pdf, equation 6. Thanks in advance

Answer 1

Yes, we can.

For example,$$\prod_{j=1}^{3}g(j)=g(1)\times g(2)\times g(3).$$

You can compare this with $$\sum_{j=1}^{3}g(j)=g(1)+g(2)+g(3).$$

Answer 2

You can, yes. But you should make the indices a proper index, i.e. $p_1 p_2 p_3$ to avoid confusion with $p\cdot1\cdot p\cdot2\cdot p\cdot 3 = 6p^3$
Also note that $p1, p2, p3$ depend on $i$. A more clear notation would give

$$f_i^{eq} = \rho\omega_i p_{i,1} p_{i,2} p_{i,3}$$

In general the product notation, just like the sum notation is quite straight-forward: $$\begin{align*} \prod_{j=1}^N p(j) &:= p(1) \cdot p(2)\cdot\ldots\cdot p(N)\\ \prod_{j=1}^\infty p(j) &:= \lim_{N\to\infty} \prod_{j=1}^N p(j) \end{align*}$$