All Questions
5
questions
6
votes
1
answer
86
views
If $r>0$ and $r\notin \mathbb{N}$, is there a simple method to evaluate $ \sum_{n=\lceil r \rceil}^{\infty} {\binom{n}{r}^{-1}}?$
Let $r>0,r\in \mathbb{R}\setminus\mathbb{N}$. Empirically, I have noticed the following relation:
$$
\sum_{n=0}^{\lfloor r \rfloor} \frac{1}{\binom{n}{r}} = - \sum_{n=\lceil r \rceil}^{\infty} \...
- 8,369
1
vote
0
answers
95
views
Is there a closed form for the binomial expression $\binom{-1/m}{k} $?
I'm interested in binomial coefficients of the form $$\binom{-1/m}{k} ,$$ where $m$ is a positive integer.
For $m=2$, it holds that \begin{align} \binom{-1/2}{k} &= (-4)^{-k} \binom{2k}{k} \qquad ...
- 7,148
2
votes
1
answer
152
views
Closed form of $\prod_{i=0}^{N}\big(i!\big)^{{N}\choose{i}}$
I was wondering if there is a closed form for
$$\prod_{i=0}^{N}\big(i!\big)^{{N}\choose{i}}$$
I know that for
$$\prod_{i=0}^{N}\big(i!\big)=G(N+2)$$
where we have expressed it as Barnes G-function. ...
- 493
1
vote
1
answer
45
views
Expressing the coefficients of $(1-x)^{1/4}$ using factorials
From the fact that $1\times3\times5\times\ldots\times(2n-1)=\frac{(2n)!}{2^nn!}$, we can show that
$$
(1-x)^{1/2}=\sum_{n=0}^\infty \frac{(2n-2)!}{(n-1)!n!2^{2n+1}}x^n.
$$
However, can I do the same ...
- 7,534
5
votes
3
answers
1k
views
Hypergeometric 2F1 with negative c
I've got this hypergeometric series
$_2F_1 \left[ \begin{array}{ll}
a &-n \\
-a-n+1 &
\end{array} ; 1\right]$
where $a,n>0$ and $a,n\in \mathbb{N}$
The problem is that $-a-n+1$ is ...
- 159