Analytically continuing the product of the first $n!$ to negative numbers?

Question

Analytically continuing the product of the first $n!$

I recently had the following idea to use the below identity:

$$ (1!2! 3! \dots n!) (12^2 3^3 4^4 \dots n^n) = n!^{n+1}$$

If we focus on the part below:

$$ P(n)=(12^2 3^3 4^4 \dots n^n) $$

We notice:

$$ P(n+1) = (n+1)^{n+1} P(n)$$

with $P(1) =1$

I noted if we define $P(n) = n^n$ for $0<n \leq 1$

Going back to the original equation:

$$ (1!2! 3! \dots n!) = \frac{\Gamma(n+1)^{n+1}}{ P(n)}$$

We notice the L.H.S is defined for integer $n$ while the R.H.S is defined for the positive real number line!

Question

Is there a method to analytically continue the product of the first $n!$ to the negative part of the number line in a smooth manner? And if we are a bit ambitious the entire complex plane?

Answer 1

See the Barnes G-function. Defined here: https://en.m.wikipedia.org/wiki/Barnes_G-function?wprov=sfla1