How do I make a $\prod$ function explicit?

Question

I've got a $\prod$ (product operator) function that I'm trying to make explicit. I've managed to convert everything else to explicit form, which we can call $g(x)$, except for this one part, so overall I've got:

$$f(x)=g(x)\prod_{n=1}^x{n^k}$$

How do I make that product term explicit? Overall, the effect I'm trying to produce is this:

$$\textrm{let} \ h(x) = \frac{f(x)}{g(x)}$$ $$h(0) = 0$$ $$h(1) = 1$$ $$h(2) = 2^k$$ $$h(3) = 2^k3^k$$ $$h(4) = 2^k3^k4^k$$ $$h(5) = 2^k3^k4^k5^k$$

Can I get a function for $h(x)$ that's not dependent on the product operator?

Answer 1

For $x\in\mathbb{N}\cup \{0\} $, we have $h(x)=(x!)^k$ which for general $x\geq 0$ (and even more general, for $x\ \in \{y\ | \ y\in\mathbb{C} \wedge \Re(y)>-1$ } ) becomes $h(x)=\Gamma(x+1)^k$ where $\Gamma(.)$ is the Gamma function. As noted in the comment, this would just be a different notation for the product operator.

Answer 2

We have $$ h(x) = \prod_{1\leq n\leq x} n^k = ([x]!)^k, $$ for $x\geq1$, where $[x]$ is the integer part of $x$.