What formula relates the factorial of a product to the product of factorials?

Question

$\text{We want } \, (a \cdot b)! = \text{some formula in terms of } (a!) \text{ and } (b!)$

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Let $F$ be a mapping from $(\mathbb{N} \times \mathbb{N})$ to $\mathbb{N}$ such that $\forall a, b \in \mathbb{N}, \quad (a\cdot b)! = F(a! \, , b!)$

Is there a nice closed-form formula for $F$ which does not directly refer to the gamma function, or inverse gamma function?

Answer 1

Using the gamma function in place of the factorial, there is Gauss multiplication formula (have a look here)

$$\Gamma(ab)=(2 \pi )^{\frac{1-a}{2}}\, a^{a b-\frac{1}{2}}\,\prod_{k=0}^{a-1}\Gamma \left(b+\frac{k}{a}\right)$$

But$$(ab)!=\Gamma(ab+1)=ab \,\Gamma(ab)$$ then $$(ab)!=(2 \pi )^{\frac{1-a}{2}}\, a^{a b+\frac{1}{2}}b\,\prod_{k=0}^{a-1}\Gamma \left(b+\frac{k}{a}\right)$$ and you can again transform the gamma function in terms of factorials.