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Working with sums/products of lists

I need to work symbolically with expressions such as this

$L(x; a,b) = \prod_{i=1}^n{abx_i^{a-1}(1-x_i^a)^{b-1}}$

where $x$ would be a random sample of size $n$.

I don't know of any way to express the indexing of the sample $x$ by each element...

The closest I got was defining the variables $n$ and $i$ and representing $x$ as a function

X = function('X',nargs=1)
L = product(a*b*X(i)^(a-1)*(1-X(i)^a)^(b-1), i, 1, n)

But this seems to inmediately assume that $X(i) = i$ and $L$ is represented as:

$-\frac{\left(-1\right)^{n} a^{n} b^{n} X\left(0\right) X\left(-1\right) X\left(-2\right) X\left(-3\right) X\left(-4\right) X\left(-5\right) X\left(-6\right) X\left(-7\right) X\left(-8\right) X\left(-9\right) {\prod_{i=1}^{n} {\left(-X\left(i\right)^{a} + 1\right)}^{b}} {\prod_{i=1}^{n} X\left(i\right)^{a}}}{X\left(n - 1\right) X\left(n - 2\right) X\left(n - 3\right) X\left(n - 4\right) X\left(n - 5\right) X\left(n - 6\right) X\left(n - 7\right) X\left(n - 8\right) X\left(n - 9\right) X\left(n\right) {\prod_{i=1}^{n} X\left(i\right)^{a} - 1} {\prod_{i=1}^{n} X\left(i - 10\right)}}$

I don't know how to deal with this expression, and it seems to me like it should be straight forward.

In case it's meaningful, after defining the expression, I will be differentiating it with respect to both $a$ and $b$.