All Questions
4
questions
17
votes
1
answer
1k
views
What is $\mathcal{R}$?
First of all, I am asking this question entirely out of curiosity. It basically randomly popped out of my mind.
So I am asking for the value of an infinite series.
Let's call it, $\mathcal{R}=\sum_{n=...
- 2,469
2
votes
1
answer
152
views
Closed form of $\prod_{i=0}^{N}\big(i!\big)^{{N}\choose{i}}$
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. ...
- 493
1
vote
0
answers
68
views
Closed form for product over Gamma function
Is there a "closed form" (with which I mean an expression not involving an indexed sum or product) for any of these four products?
$$\prod_{k=1}^{n} \Gamma(\frac{x}{k*2+1})$$
$$\prod_{k=1}^{n} \Gamma(...
4
votes
1
answer
182
views
Product identity for $n^n$
I came across the rather nice identity
\begin{align}
&&\frac{(-n)^{n-1} \Gamma (n+1)}{(1-n)_{n-1}}&&\tag{1}&\\
\\
&=&\prod _{k=1}^{n-1} \frac{(k+1) n^2}{n^2-k n}&&\...
- 9,018