All Questions
19
questions
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
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,
$$\frac{\Gamma\left(\frac{1}{24}\right)\Gamma\left(\...
- 667
3
votes
1
answer
125
views
Neat function $I(x,y)= \sum\limits_{n=1}^{\infty}\bigg|\int_0^1 \frac{1}{t~(\log t)^y}\exp\bigg(\frac{n^x}{\log t}\bigg) ~dt~\bigg| $. Closed form?
Consider the following function:
$$ 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|$$
For $x=0$ and letting $y$ vary we get the Gamma ...
- 756
15
votes
3
answers
1k
views
Closed-form of log gamma integral $\int_0^z\ln\Gamma(t)~dt$ for $z =1,\frac12, \frac13, \frac14, \frac16,$ using Catalan's and Gieseking's constant?
We have the known,
$$I(z)=\int_0^z\ln\Gamma(t)~dt=\frac{z(1-z)}2+\frac z2\ln(2\pi)+z\ln\Gamma(z)-\ln G(z+1)$$
or alternatively,
$$I(z)=\int_0^z\ln\Gamma(t)~dt= \frac{z(1-z)}{2}+\frac{z}{2}\ln(2\pi) -(...
- 54.9k
1
vote
0
answers
89
views
Yet another bizarre identity involving hypergoemetric functions and gamma functions.
Let $d=4$, $T\ge d$ and $p\ge 0$ be integers.
By solving Spectral densities of finite dimensional sample covariance matrices we stumbled on a following identity.
\begin{eqnarray}
&&\sum\...
- 11.5k
3
votes
0
answers
84
views
Closed form for infinite series $\sum_{n=1}^\infty \prod_{j=1}^n \left[1-(\tfrac{j}{n}u+v)^2\right]^{-1} x^n$
I've been struggling to find a closed form for the following series that depends on $u$ and $v$ as parameters:
$$
f(u,v,x) = \sum_{n=1}^\infty c_n x^n
$$
with
$$
c_n = \frac{1}{\prod_{j=1}^n \left[...
- 207
6
votes
4
answers
531
views
Prove that $2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)} \, \mathrm{d}x =\ln(\pi)-\gamma $
Let $\gamma$ be the Euler-Mascheroni constant.
I'm trying to prove that
$$2\int_0^\infty \frac{e^x-x-1}{x(e^{2x}-1)} \, \mathrm{d}x =\ln(\pi)-\gamma $$
I tried introducing a parameter to the ...
user334593
4
votes
1
answer
215
views
How to prove $H\left(\frac{1}{4}\right)=\frac{e^{\frac{C}{4\pi}-\frac{3}{32}}\cdot A^{\frac{9}{8}}}{\sqrt 2}$
How to prove:
$$
H\left(\frac{1}{4}\right)=\frac{e^{\frac{C}{4\pi}-\frac{3}{32}}\cdot A^{\frac{9}{8}}}{\sqrt 2}
$$
Where $C$ is Catalan's number, $A$ is Glaisher-Kinkelin's constant and $H(x)$ is the ...
- 12.1k
15
votes
1
answer
599
views
Closed-form of $\int_0^1 \left(\ln \Gamma(x)\right)^3\,dx$
From the amazing result by Raabe we know that
$$LG_1=\int_0^1 \ln \Gamma(x)\,dx = \frac{1}{2}\ln(2\pi) = -\zeta'(0).$$
We also know that
$$LG_2 = \int_0^1 \left(\ln \Gamma(x)\right)^2\,dx = \frac{\...
- 12.4k
4
votes
0
answers
281
views
Non-recursive closed-form of the coefficients of Taylor series of the reciprocal gamma function
The reciprocal gamma function has the following Taylor series.
$$\frac{1}{\Gamma(z)}=\sum_{k=1}^{\infty}a_kz^k,$$
where the $a_k$ coefficient are given by the followint recursion.
$a_1=1$, $a_2=\...
- 12.4k
3
votes
1
answer
371
views
Closed form for this incomplete gamma series?
The series I'm working with is
$$\sum_{k=0}^\infty \binom{z}{k}(-1)^k ( 1-\frac{\Gamma(k,-\log n)}{\Gamma(k)} )$$
with $z$ a complex variable and $\Gamma(k, -\log n)$ the upper incomplete gamma ...
- 987
7
votes
3
answers
518
views
A closed-form of product the gamma functions containing $\pi$ and $\phi$
Playing with gamma functions by randomly inputting numbers to Wolfram Alpha, I got the following beautiful result
\begin{equation}
\frac{\Gamma\left(\frac{3}{10}\right)\Gamma\left(\frac{4}{10}\...
- 19.4k
25
votes
1
answer
1k
views
Strange closed forms for hypergeometric functions
So in the process of trying to find a derivation for this answer, the following interesting equalities arose (one can check with Wolfram Alpha/Mathematica):
$$\frac{8\sqrt{2}G^4}{5\pi^2} \left(\left(...
- 1,827
0
votes
2
answers
271
views
Expressing $\mathrm{B}(\sinh(x), \cosh(x))$ in terms of elementary functions
Is it possible to express:
$\mathrm{B}(\sinh(x), \cosh(x))$
(where $\mathrm{B}$ is the beta function)
In closed form, in terms of elementary functions?
- 2,323
60
votes
1
answer
6k
views
Is it possible to simplify $\frac{\Gamma\left(\frac{1}{10}\right)}{\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$?
Is it possible to simplify this expression?
$$\frac{\displaystyle\Gamma\left(\frac{1}{10}\right)}{\displaystyle\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$$
Is there a systematic ...
- 2,602