All Questions
6
questions
2
votes
0
answers
52
views
How to define double factorial for non positive integers?
I studied double factorial which known for natural number
$$ n!!=n(n-2)!! , 1!!=0!!=1$$
So we have for $n\in N$
$$ (2n)!!=2^n n! , (2n+1)!!=\frac{(2n+1)!}{2^n n!}$$
but I found on Math-World formula ...
- 1,637
5
votes
1
answer
231
views
Closed form for $\Gamma(a-x)$ where $a \in (0,1]$.
Now asked on MO here.
I wonder if there is a closed form for $ \Gamma(a-x)$.
And by closed form here I mean a finite combinations of elementary functions, powers of $\Gamma(a)$ and powers of $\Gamma(...
- 6,620
3
votes
0
answers
53
views
Find $\prod_{k=1}^n \frac{\Gamma (a_k/m)}{\Gamma (b_k/m)}$ algorithmically
It sometimes happens that
$$\prod_{k=1}^n \frac{\Gamma (a_k/m)}{\Gamma (b_k/m)}$$
is algebraic for positive integers $m,n,a_k,b_k$. For example,
$$\frac{\Gamma\left(\frac{1}{24}\right)\Gamma\left(\...
- 667
4
votes
0
answers
96
views
Evaluate $\sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(2b)_n} \frac{\psi^{(0)}\left (n+ \frac{a}2+1 \right ) }{\Gamma\left (n+ \frac{a}2+1 \right ) }$
Define
$$S(a,b)=\sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(2b)_n} \frac{\psi^{(0)}\left (n+ \frac{a}2+1 \right ) }{\Gamma\left (n+ \frac{a}2+1 \right ) },$$
where $\psi^{(0)}(x)=\frac{\Gamma^\prime(x)}{\...
- 5,370
48
votes
1
answer
1k
views
How to evaluate double limit of multifactorial $\lim\limits_{k\to\infty}\lim\limits_{n\to 0} \sqrt[n]{n\underbrace{!!!!\cdots!}_{k\,\text{times}}}$
Define the multifactorial function $$n!^{(k)}=n(n-k)(n-2k)\cdots$$ where the product extends to the least positive integer of $n$ modulo $k$. In this answer, I derived one of several analytic ...
- 27.1k
2
votes
1
answer
76
views
How to compute: $I_x =\int_0^1 t^x e^{-\frac{1}{1-t^2}} dt~~~x>0$?
$$I_x =\int_0^1 t^x e^{-\frac{1}{1-t^2}} dt~~~x>0$$
I first tried to see if I could solve the integer case.
$$I_n =\int_0^1 t^ne^{-\frac{1}{1-t^2}} dt $$
I have tried to find a possible ...
- 24.2k