All Questions
Tagged with closed-form gamma-function
109
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
0
votes
0
answers
55
views
how to use Gauss Multiplication Formula for Gamma function?
I studied Gauss Multiplication Formula which known for $n\in Z^+ \wedge nx\notin Z^-\cup\{0\}$
$$\Gamma(nz)=(2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\Gamma\left(z+\frac{k}{n}\right)$$
but I didn't ...
- 1,637
3
votes
2
answers
114
views
Is it possible to find the $n$th derivative of Gamma function?
By repeatedly differentiating $\Gamma(x)$, I noticed that
$$\frac{d^{n}}{{dx}^{n}}\Gamma(x)=\sum_{k=0}^{n-1}\binom{n-1}{k}\psi^{(n-k-1)}(x)\,\frac{d^{k}}{{dx}^{k}}\Gamma(x),$$
where $\psi^{(a)}(x)$ is ...
- 25.8k
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
2
votes
0
answers
238
views
Showing $\int_{0}^{1}\frac{E(\tfrac{x}{\sqrt{x^2+8}})}{\sqrt{8-7x^2-x^4}}dx=\frac{1}{3}K(\frac{1}{\sqrt{2}})E(\frac{1}{\sqrt{2}})$
Context
$\begin{align}
K(k)=\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-k^2\sin^2t}}\tag{1}
\end{align}$
and
$\begin{align}
E(k)=\int_{0}^{\pi/2}\sqrt{1-k^2\sin^2t}dt\tag{2}
\end{align}$
the complete elliptic ...
- 323
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,
$$\frac{\Gamma\left(\frac{1}{24}\right)\Gamma\left(\...
- 667
0
votes
0
answers
74
views
Is it possible to find a closed form for $i!$? [duplicate]
I am curious is there a closed form for $i!$? I tried to search for any closed form for this but I didn't find any.
$$z! := \lim_{n \to \infty } n^z \prod_{k=1}^n \frac {k}{z+k}$$
$$i! =\lim_{n \to \...
2
votes
1
answer
164
views
Another weird limit involving gamma and digamma function via continued fraction
Context :
I want to find a closed form to :
$$\lim_{x\to 0}\left(\frac{f(x)}{f(0)}\right)^{\frac{1}{x}}=L,f(x)=\left(\frac{1}{1+x}\right)!×\left(\frac{1}{1+\frac{1}{1+x}}\right)!\cdots$$
Some ...
4
votes
2
answers
188
views
A closed form for integrals of the type $\Gamma\left(\sigma+it\right)$?
Numerical evidence strongly suggests that:
$$\int_{-\infty}^{\infty}\Gamma\left(\sigma+it\right) \,\mathrm{d}t = 2\cdot\frac{\pi}{\mathrm{e}} \qquad \sigma \in \mathbb{R}, \sigma > 0$$
and
$$\int_{-...
- 3,191
3
votes
1
answer
163
views
Does a closed-form expression exist for $ \int_0^\infty \ln(x) \operatorname{sech}(x)^n dx $?
I am trying to find a closed-form expression for the following integral
$$
\int_0^\infty \ln(x) \operatorname{sech}(x)^n dx
$$
There are specific values that I would like to generate
(Table of ...
7
votes
0
answers
194
views
The closed-form of $1-5\left(\frac{1}{2}\right)^k+9\left(\frac{1\cdot3}{2\cdot4}\right)^k-13\left(\frac{1\cdot3\cdot5}{2\cdot4\cdot6}\right)^k+\dots$?
(A related MSE question by P. Singh.) First define,
$$F_k = 1-5\left(\frac{1}{2}\right)^k+9\left(\frac{1\cdot 3}{2\cdot 4}\right)^k-13\left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right)^k+17\left(\...
- 54.9k
1
vote
0
answers
52
views
Does the rest of this family of continued fractions have closed forms?
The pattern for the continued fractions below is quite straightforward. $F_1$ has numerators with all the integers but,
$F_2\; \text{is missing}\; 2m+1 = 3,5,7,\dots\\
F_3\; \text{is missing}\; 3m+1 = ...
- 54.9k
3
votes
2
answers
191
views
Proving $\sum_{n=-\infty}^\infty n^2e^{-\pi n^2}=\frac{\Gamma (1/4)}{4\sqrt{2}\pi^{7/4}}$
I conjecture that
$$\sum_{n=-\infty}^\infty n^2e^{-\pi n^2}=\frac{\Gamma (1/4)}{4\sqrt{2}\pi^{7/4}}$$
because the left-hand side and right-hand side agree to at least $50$ decimal places. Is the ...
- 667
9
votes
2
answers
341
views
On the cubic counterpart of Ramanujan's $\sqrt{\frac{\pi\,e}{2}} =1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots$?
We have Ramanujan's well-known,
$$\sqrt{\frac{\pi\,e}{2}}
=1+\frac{1}{1\cdot3}+\frac{1}{1\cdot3\cdot5}+\frac{1}{1\cdot3\cdot5\cdot7}+\dots\color{blue}+\,\cfrac1{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{3}{1+\...
- 54.9k
0
votes
0
answers
34
views
Does the following expression have a closed form?
I have a ratio that has upper incomplete Gamma functions.
$$ r = \frac{\Gamma(n, (a+c) x) - \Gamma(n, (a+c) y)}{\Gamma(n, a x) - \Gamma(n, a y)} $$
Here, $a, c, x, y$, are positive real and $y > x$....
- 109
4
votes
0
answers
96
views
Evaluate $\sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(2b)_n} \frac{\psi^{(0)}\left (n+ \frac{a}2+1 \right ) }{\Gamma\left (n+ \frac{a}2+1 \right ) }$
Define
$$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 ) },$$
where $\psi^{(0)}(x)=\frac{\Gamma^\prime(x)}{\...
- 5,370
2
votes
1
answer
115
views
Generalization of Gauss multiplication formula for $\Gamma(jm+kn+a);j,k\in\Bbb N$?
A hypergeometric single sum, like a Mittag Leffler function uses the Pochhammer symbol $(a)_n$ multiplication formula to easily have a univariate hypergeometric function $_p\text F_q$ closed form:
$$\...
- 12.5k
1
vote
1
answer
61
views
Closed form for $\int_0^{\pi/2}(\cos^2{\theta})^{1/n}d\theta$
Is there a closed form for $f(n)=\int_0^{\pi/2}(\cos^2{\theta})^{1/n}d\theta$ for every $n\in\mathbb{N}$ ?
I suspect there may be, because of the following apparent equalities.
$f(1)=\frac{\pi}{4}$
$f(...
- 25.7k
1
vote
1
answer
133
views
Does $\lim_{x\to 0} \left(2^{1-x!}3^{1-x!!}4^{1-x!!!}5^{1-x!!!!}6^{1-x!!!!!}\cdot\cdot\cdot\right)^{\frac{1}{x}}=L$ admits a closed form?
I try to simplify this limit :
$$\lim_{x\to 0} \left(2^{1-x!}3^{1-x!!}4^{1-x!!!}5^{1-x!!!!}6^{1-x!!!!!}\cdots\right)^{\frac{1}{x}}=L$$
Where we compose the Gamma function with itself .
From the past ...
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
0
votes
1
answer
54
views
Find $a$ such that the limit is zero
Problem :
Let $x>0$ then define :
$$f(x)=\left(\left(\frac{1}{x}\right)!\left(x!\right)\right)^{\frac{1}{x+\frac{1}{x}}}$$
Then find $a$ such that :
$$\lim_{x\to\infty}f(x)-\frac{1}{2}\left(\frac{1}...
3
votes
1
answer
125
views
Neat function $I(x,y)= \sum\limits_{n=1}^{\infty}\bigg|\int_0^1 \frac{1}{t~(\log t)^y}\exp\bigg(\frac{n^x}{\log t}\bigg) ~dt~\bigg| $. Closed form?
Consider the following function:
$$ I(x,y)=\sum_{n=1}^{\infty}\bigg|\int_0^1 \frac{1}{t~(\log t)^y}\exp\bigg(\frac{n^x}{\log t}\bigg) ~dt~\bigg|$$
For $x=0$ and letting $y$ vary we get the Gamma ...
- 756
3
votes
0
answers
173
views
Question on a closed-form expression related to the harmonic number $H_n$
In this question the notation $\tilde{f}(x)$ refers to an analytic representation of the summatory function
$$f(x)=\sum\limits_{n=1}^x a(n)\tag{1}$$
that converges to
$$\underset{\epsilon\to 0}{\text{...
- 7,631
0
votes
2
answers
145
views
Closed from for the series involving gamma function
Is there a closed form for the fallowing series,
$$\sum_{n=1}^\infty \Gamma\left(n+\frac12+\frac12k\right)\Gamma\left(n+\frac12-\frac12k\right)\frac{x^{n}}{(2n+1)!}$$
where $k\notin\mathbb{Z}$.
I ...
user965146
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
6
votes
1
answer
307
views
Closed form of $\sum_{n=1}^\infty \frac{1}{\sinh n\pi}$ in terms of $\Gamma (a)$, $a\in\mathbb{Q}$
This question and this question are about
$$\sum_{n=1}^\infty \frac{1}{\cosh n\pi}=\frac{1}{2}\left(\frac{\sqrt{\pi}}{\Gamma ^2(3/4)}-1\right)$$
and
$$\sum_{n=1}^\infty \frac{1}{\sinh ^2n\pi}=\frac{1}{...
- 966
1
vote
1
answer
123
views
Proof about the power series of reciprocal multifactorials $m_x(k)=\sum_{n=0}^\infty \frac{x^n}{n\underbrace{!\cdots!}_{\text{k times}}}$
The proof I've attempted mimics very closely the answer on this question.
How to prove the formula for the Reciprocal Multifactorial constant?
Pre-requisite definitions:
A multifactorial of order $k \...
- 407
0
votes
0
answers
41
views
Incomplete upper gamma for a non-integer number of degrees of freedom
I can't seem to nail the closed form of the incomplete upper gamma function for the number of degrees of freedom $s$ being a fraction type $n/2$ where $n$ is integer. For the case when $s$ is integer ...
48
votes
1
answer
1k
views
How to evaluate double limit of multifactorial $\lim\limits_{k\to\infty}\lim\limits_{n\to 0} \sqrt[n]{n\underbrace{!!!!\cdots!}_{k\,\text{times}}}$
Define the multifactorial function $$n!^{(k)}=n(n-k)(n-2k)\cdots$$ where the product extends to the least positive integer of $n$ modulo $k$. In this answer, I derived one of several analytic ...
- 27.1k
6
votes
1
answer
86
views
If $r>0$ and $r\notin \mathbb{N}$, is there a simple method to evaluate $ \sum_{n=\lceil r \rceil}^{\infty} {\binom{n}{r}^{-1}}?$
Let $r>0,r\in \mathbb{R}\setminus\mathbb{N}$. Empirically, I have noticed the following relation:
$$
\sum_{n=0}^{\lfloor r \rfloor} \frac{1}{\binom{n}{r}} = - \sum_{n=\lceil r \rceil}^{\infty} \...
- 8,369