All Questions
11
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
0
votes
0
answers
74
views
Is it possible to find a closed form for $i!$? [duplicate]
I am curious is there a closed form for $i!$? I tried to search for any closed form for this but I didn't find any.
$$z! := \lim_{n \to \infty } n^z \prod_{k=1}^n \frac {k}{z+k}$$
$$i! =\lim_{n \to \...
1
vote
0
answers
43
views
Computing $\frac{d^{n-1}}{dz^{n-1}}\frac{z}{\ln\left(z!\right)}^{n}$
I am trying to compute
$$\frac{d^{n-1}}{dz^{n-1}}\left(\frac{z}{\ln\left(z!\right)}\right)^{n}$$
The problem arises when dealing with inversion formulae. My question is, can this expression be ...
- 11
1
vote
1
answer
123
views
Proof about the power series of reciprocal multifactorials $m_x(k)=\sum_{n=0}^\infty \frac{x^n}{n\underbrace{!\cdots!}_{\text{k times}}}$
The proof I've attempted mimics very closely the answer on this question.
How to prove the formula for the Reciprocal Multifactorial constant?
Pre-requisite definitions:
A multifactorial of order $k \...
- 407
48
votes
1
answer
1k
views
How to evaluate double limit of multifactorial $\lim\limits_{k\to\infty}\lim\limits_{n\to 0} \sqrt[n]{n\underbrace{!!!!\cdots!}_{k\,\text{times}}}$
Define the multifactorial function $$n!^{(k)}=n(n-k)(n-2k)\cdots$$ where the product extends to the least positive integer of $n$ modulo $k$. In this answer, I derived one of several analytic ...
- 27.1k
1
vote
1
answer
45
views
Expressing the coefficients of $(1-x)^{1/4}$ using factorials
From the fact that $1\times3\times5\times\ldots\times(2n-1)=\frac{(2n)!}{2^nn!}$, we can show that
$$
(1-x)^{1/2}=\sum_{n=0}^\infty \frac{(2n-2)!}{(n-1)!n!2^{2n+1}}x^n.
$$
However, can I do the same ...
- 7,534
2
votes
3
answers
130
views
Does this function have a closed form?
$$k_a(x)=\sum^\infty_{n=0}{\frac{n^2}{(n+x)!}}$$
$$k_b(x)=\sum^\infty_{n=0}{\frac{(n+x)^2}{n!}}$$
I noticed these functions closely relate to $e$. By looking at them I was able to determain a closed ...
- 4,472
2
votes
1
answer
232
views
Approximation of the factorial function
I'm using the term factorial function as $\gamma(x+1)$ on the sense that I'm taking all real number in count.
I have seen many approximations of the factorial function for positive values, for ...
- 5,166
2
votes
2
answers
163
views
Equations involving factorial/Gamma function
Are there any known methods to formally solve equations like:
1)$x^3!+(2x^2)!-x!+3=0$
2)$x!=e^x$ ($0$ is trivial but there must be another one)
3)$(2x!)^2+x!-1=0$
4)$x!!+x!=7$
I don't need ...
- 4,099
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