sorry if this was already posted, consider the product involving $n$ variables $X_i$, $$ P=\prod_{i=1}^n X_i^{k_i}(1-X_i)^{1-k_i}, $$ where $k_i$ ($i=1,...,n$) assumes either 0 or 1. Thinking of all the combinations of $(k_1,\cdots,k_n)$ of $n$ binary terms, can we convert the product above in a summation for any sequence of binary terms? I mean, without log-transformation.
My try: $$ X_1^{k_1}(1-X_1)^{1-k_1}\times X_2^{k_2}(1-X_2)^{1-k_2}\times \cdots\times X_n^{k_n}(1-X_n)^{1-k_n}, $$ for the sequence ($0,\cdots,0$) $$ (1-X_1)\times (1-X_2)\times \cdots\times (1-X_n), $$ which may be expressed through multi-binomial theorem, such as in Binomial product expansion (I would like to have something similar)
However, as the complexity of the combinations increases (that is, many 1's and 0's), I find a bit difficult to find a general expression. I'd appreciate any suggestion. Thanks in advance.
Hye,
Denoting $I_0=\{i\mid k_i=0\}$ and $I_1=\{i\mid k_i=1\}$, you can develop your polynomial as follows:
$$\left(\prod_{i\in I_1}X_i\right)\left(\prod_{i\in I_0}(1-X_i)\right)=\left(\prod_{i\in I_1}X_i\right)\left(\sum_{I\subset I_0}(-1)^{|I|}\prod_{i\in I}X_i\right)=(-1)^{|I_1|}\sum_{I\mid I_1\subset I}(-1)^{|I|}\prod_{i\in I}X_i.$$
I hope this can help.