Closed form of $\prod_{i=0}^{N}\big(i!\big)^{{N}\choose{i}}$

Question

I was wondering if there is a closed form for $$\prod_{i=0}^{N}\big(i!\big)^{{N}\choose{i}}$$ I know that for $$\prod_{i=0}^{N}\big(i!\big)=G(N+2)$$ where we have expressed it as Barnes G-function. Yet I am not sure how to go about this with the binomial involved?

Answer 1

An asymptotic formula can be derived that is better than 0.1% accurate for $n>7.$ I'll work with the logarithm and sketch a proof that

$$ (*) \quad S_n :=\sum_{k=0}^n \binom{n}{k} \log \Gamma(k+1) \sim 2^n\Big( \log \Gamma(n/2+1) + 1/4 - 1/(8n) + ... \Big) $$

Use the following formula (it can be found on the wiki for $\Gamma$ ) $$ \log \Gamma(k+1) = -\gamma \, k + \int_0^\infty \frac{e^{-k\,t} -1 + k \,t}{e^t-1} \frac{dt}{t} $$

Insert the previous equation into the left-hand side of (*), switch the finite sum with the integral, do the sum, and factor out a $2^n,$

$$ S_n = 2^n\Big(-\gamma \frac{n}{2} + \int_0^\infty \frac{e^{-n\,t/2}\cosh^n(t/2) -1 + \frac{n}{2} \,t}{e^t-1} \frac{dt}{t} \Big)$$ Note I've used a hyperbolic trig ID. The reason is because, for $t$ small enough, $$ \cosh^n(t/2) = 1 + \frac{n\,t^2}{8} + \frac{(3n-2)n\,t^4}{384}+...$$ Therefore, $$ S_n \sim 2^n\Big( -\gamma \frac{n}{2} + \int_0^\infty \frac{e^{-n\,t /2} -1 + \frac{n}{2} \,t}{e^t-1} \frac{dt}{t} +\int_0^\infty \frac{dt}{t} \frac{e^{-n\,t/2 }}{e^t-1} \big(\frac{n\,t^2}{8} + \frac{(3n-2)n\,t^4}{384} \big) \Big)$$ $$ = 2^n\Big(\log\Gamma(1+n/2)+\frac{n}{8}\int_0^\infty \frac{e^{-n\,t/2 }}{e^t-1}t\,dt + \frac{(3n-2)n}{384}\int_0^\infty \frac{e^{-n\,t/2 }}{e^t-1}t^3\,dt + ... \Big)$$

We can split the integral up because each converges (the $t$ is denominator is cancelled). More importantly the terms shown, and the subsequent ones, form an asymptotic sequence in $n$. In particular, the integrals can be explicitly written in terms of odd derivatives of the polygamma function, and since the polygamma function asymptotics is well-known, you can derive (by hand, if you want) the terms given in (*).

The remark about the accuracy of the expansion is from some numerical calculations. Take the $LHS(*)/RHS(*)$ and for n=7 the ratio is 0.99949 and for n=20, the ratio is 0.999993

Incidentally, the asymptotics proves that the product form will not be a Barnes G-function.