All Questions
Tagged with closed-form gamma-function
109
questions
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