Questions tagged [gamma-function]
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3
votes
0answers
80 views
Gosper's Beta function identities
According to the last paragraph in Mathworld's Beta function article http://mathworld.wolfram.com/BetaFunction.html, Gosper found some multiplication formulas for the Beta function, but it does not ...
14
votes
2answers
327 views
A tantalizing Gamma quotient to challenge the Rohrlich-Lang Conjecture
The Rohrlich-Lang Conjecture for polynomial relations in Gamma values predicts that all polynomial relations between Gamma values over $\mathbb Q$ come from the functional equations satisfied by the ...
2
votes
1answer
198 views
Complicated bound after using Stirling's approximation
I have this inequality $$\frac{1}{a}\exp\bigl\{-\frac{4}{h^2}\bigr\} \geq \frac{1}{f}$$ where $$ a \leq \Bigl(\pi^{d/2}\Gamma(\frac{1}{2}d+1)^{-1} + 1\Bigr) \left(\frac{h^{d+1}}{2} \Gamma \left(\frac{...
1
vote
1answer
124 views
Upper bound of the fraction of gamma functions
Is there a simple upper bound of the following fraction of gamma functions for any $a,b\geq1/2$:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}$$
An upper bound in the following form is ...
1
vote
0answers
100 views
Brylinski Beta Function Calculation
I've recently read the paper written by Brylinski on the Beta function of a knot, where he gave the example of a trivial "circular" knot. Having a physics background, and not being formally introduced ...
1
vote
3answers
148 views
Estimate for the binomial coefficients and bounds from below for the Beta function
Let $n\ge p\in \mathbb N$ and let $\binom{n}{p}$ be the binomial coefficient. I believe that
$$
\binom{n}{p}\le 2^n\sqrt\frac{2}{π n}.
$$
Question: is that true? Of course I would like it as a non-...
2
votes
1answer
107 views
Monotonicity of the regularized incomplete gamma function
In the theory of chi-squared distribution in statistics, for the random variable $X$ following $\chi^2 (k)$, the probability that $X$ is lower than its expactation is given by
$$
P(X\le k) = \frac{\...
5
votes
0answers
91 views
Relation between Gross-Koblitz and Chowla-Selberg formulas
The Chowla-Selberg formula relates the eta function with values of the gamma function at rational numbers. The eta function appears, at least in the proofs I have seen, related to values of $L$-...
1
vote
1answer
456 views
Series involving factorials
Playing around with this series for natural values of $a,b$, it appears that more generally for $c\in\mathbb N$, $$\sum_{k=0}^\infty \frac{ (a+k)! \ (b+k)!}{k!\ (a+b+c+ k+1)! }=\frac{a!\ b!\ (c-1)!}{...
3
votes
2answers
370 views
Solution of the functional equation $f(x+1)=g(x)f(x)$
In the paper (http://www.sciencedirect.com/science/article/pii/S0022247X97953439), Webster obtained a unique solution of the functional equation $f(x+1)=g(x)f(x)$
(where $f,g:\mathbb{R}^+\rightarrow \...
1
vote
0answers
106 views
The relationship between $\Gamma (a)$ and $\Gamma (a+\frac{1}{2})$ [closed]
Does anybody know the relationship between the $\Gamma (a)$ and $\Gamma (a+\frac{1}{2})$? Is there an equation or an approximate equality between these two?
1
vote
0answers
73 views
Generalized hypergeometric function at $z=1$
I wonder if there is a closed formula for the following generalized hypergeometric function at $z=1$:
$${}_4F_3\left(a,a,a,a;a+1,a+1,2a;1\right)$$
Specifically, I would like to have a formula in ...
3
votes
1answer
180 views
A generalized logarithmic function
Consider the function
$$f_\epsilon(x):=\int_0^1 \frac{(1-z)^x-1}{\log(1-\epsilon z)}\, dz,$$
defined for a parameter $\epsilon \in (0,1]$ and $x \geq 0$. When $\epsilon=1$, this is $\log(1+x)$. One ...
7
votes
0answers
168 views
The inverse of the digamma function
The gamma function is increases on the interval $(x_0, \infty),$ where $x_0$ denotes the unique zero of the digamma function on the positive half line.
The inverse function of gamma function defined ...
1
vote
2answers
471 views
Intuition behind the Riemann $\zeta$ functional equation
Let $\Lambda(s)=\pi^{-s/2} \Gamma\left(\frac{s}{2}\right)\zeta(s).$ Then $\Lambda\left(\frac{1}{2} + s\right) = \Lambda\left(\frac{1}{2} - s\right)$ constitutes the functional equation for the Riemann ...
5
votes
3answers
425 views
Asymptotic behavior of integral with gamma functions
Consider the following function defined for complex numbers $z\in\mathbb{C}$ with $\Re(z)\geq \frac{1}{2}$:
$$F(z)=\frac{1}{5^{\Re(z)}}\int_0^\infty \left| \frac{\Gamma(z+ix)\Gamma(z-ix)}{\Gamma(z)^2}...
2
votes
0answers
124 views
Reflection Formulas for the $\Gamma$ Function
We have
$$\begin{align}
&\Gamma\Big(1~+~x\Big)~\cdot~\Gamma\Big(1-x\Big)~=~\frac{\pi x}{\sin\pi x}
\\\\
&\Gamma\Big(\tfrac12+x\Big)~\cdot~\Gamma\Big(\tfrac12-x\Big)~=~\frac\pi{\cos\pi x}
\\\...
4
votes
2answers
208 views
An integral identity relate to the Gamma function or the Beta function
I encountered the following identity in a paper on number theory,
$$\int_{-\infty}^{\infty}\frac{dW}{(W+i)^{\frac{3}{2}}(W^2+1)^s}=\frac{e^\frac{-3\pi i}{4}\sqrt{2}\pi \Gamma(2s+\frac{1}{2})}{2^{2s}\...
9
votes
1answer
352 views
“unexpected” residue formula for $\Gamma^3(s)/(\Gamma(3s)(e^{2\pi is}-1)) $
There is a related problem in my current work: to find the residue of the following function at any negative integer $s=-n$:
$$f(s)=\frac{\Gamma^3(s)}{\Gamma(3s)(e^{2\pi is}-1)}$$
It seems to be a ...
2
votes
0answers
324 views
About the Stirling's approximation of the incomplete gamma function $\left|\gamma\left(a+ib,z\right)\right|$
Let $a+ib$ be a complex number. It is well-known that $$\left|\Gamma\left(a+ib\right)\right|\sim\sqrt{2\pi}e^{-\pi\left|b\right|/2}\left|b\right|^{a-1/2}$$ for any fixed $a$ and $\left|b\right|\...
1
vote
1answer
70 views
Independence of Gamma and Beta random variables with common term
Given $\textbf{P}$ independent and identically distributed random variables, $X_1, X_2, ..., X_P \sim \Gamma(M,2c)$ how can we prove that:
$$U = X_1 + X_2 + ... + X_P$$
and
$$V = \frac{X_1}{X_1 + ...
1
vote
0answers
282 views
Generalization of Trigonometric, Hyperbolic, $\Gamma$ and $\zeta$ Functions
It is becoming increasingly clear that the expression $~\displaystyle\prod_{n\in\mathbf M}\left[1-\frac{z^s}{(n-a)^s}\right],~$ with $~|\mathbf M|=\aleph_0,~$
$a\not\in\mathbf M,~$ and $~\Re(s)>1,...
5
votes
0answers
285 views
integrating with respect to parameters in beta function
I would like to evaluate an integral:
$$\int_t^1\frac{1}{B(1+s\phi,1+\phi(1-s))}p^{s\phi}(1-p)^{\phi(1-s)}ds,$$
where $B(a,b)$ is a beta function and $p\in(0,1)$ and $\phi>0$ are some parameters. ...
1
vote
1answer
252 views
Is this a valid way of generating the non-trivial zeros of $\zeta(s)$?
This question is migrated from this one on MSE and rephrased more simply.
In this question the following closed form was derived, with $0 < \sigma<1$ and $\sigma,x \in \mathbb{R}$:
$$ \frac{1}...
-1
votes
1answer
126 views
Closed form for sum involving digamma? [closed]
Let $\Gamma(n)$ be Euler's Gamma function and $\psi_0$ = $\frac{\Gamma'(n)}{\Gamma(n)}$ be the Digamma function.
Is there a closed form for
$$\sum_{n=1}^{\infty} \frac{\psi_0(n)}{n^2}=?$$
I've done ...
2
votes
0answers
86 views
Gamma function in terms of a linear function
I have noticed that the plot of the function $[\Gamma(x+1)]^{1/x}$ for $x > 0$ looks like a straight line at all scales. This implies that $\Gamma(x+1) \approx ((1-e^{-\gamma})x+e^{-\gamma})^x$ for ...
2
votes
1answer
178 views
How to compute the following integral $I_{\alpha,\beta}$
We have the following identity (see Bateman, H. (1953). Higher Transcendental Functions [Volumes I], p. 25.)
$$(*)\quad \Gamma(\mu)\, \zeta(\mu,\nu) = \int_{0}^{1} x^{\nu-1} \,(1-x)^{-1} \Bigr(\log ...
2
votes
0answers
179 views
Are all zeros of $\dfrac{\Gamma'}{\Gamma^2}(s) \pm \dfrac{\Gamma'}{\Gamma^2}(1-s)$ either real or on the line with $\Re(s)=\frac12$?
In this post it was proven that, except for a finite few, all zeros of $\Gamma(s) \pm \Gamma(1-s)$ are either real or reside on the line $\Re(s)=\frac12$.
I have been searching for similar reflexive $...
4
votes
1answer
295 views
Xi Function on Critical Strip - Mellin Transform
Story
I'm trying to prove following identity
$$\int_0^\infty \frac{\Xi(t)}{t^2 + \frac{1}{4}} \cos(xt) dt = \frac{1}{2} \pi (e^{\frac{1}{2}x} - 2e^{-\frac{1}{2}x} \psi(e^{-2x}))$$
where
$$\psi(x)=\...
2
votes
2answers
515 views
An integral identity evaluating the gamma function
While reading a number theory paper I encountered the identity
$$ \int_{- \infty}^{\infty} (1 + x^2)^{ - \frac{z}{2} - 1} dx = \sqrt{\pi} \frac{ \Gamma(\frac{z + 1}{2}) }{\Gamma(\frac{z}{2} + 1)},$$
...
1
vote
0answers
58 views
The integral of $\Gamma\left(\zeta\right) \, W_{-\zeta,\mu}(z) $
Someone has a reference that addresses an integral of the followns type
$$I_{a,b,x} = \int_{0}^{+\infty} \zeta^{-a} \, \Gamma\left(\zeta+b\right) \, W_{-\zeta-b,\tfrac{-1}{2}}(x) \, d\zeta$$
where ...
2
votes
2answers
85 views
Solution of $\left(\Gamma(x+c)/ \Gamma(x+d)\right)y^d/y^c = {\rm const}$
When I try to solve $F(x,y)= \Gamma(x+c)/\Gamma(x+d) y^d/y^c = {\rm const}$,
I find that $y = p x + q$ satisfies the above equation, whith specific $p$ and $q$ constants for the given constants $c$ ...
3
votes
1answer
249 views
Proof of Euler's reflection formula via rapidly decreasing Fourier series
Story
I want to prove Euler's reflection formula by showing that
\begin{equation*}
f(s) = \sin(\pi s) \Gamma(s) \Gamma(1 - s)
\end{equation*}
is constant, where $s = \sigma + it$. It's easy to see ...
2
votes
1answer
97 views
Estimating $\left(\Gamma\left(\frac{\alpha}{\sqrt n}\right)\right)^n$ for fixed $\alpha >0$ as a function of (large) $n$ [closed]
I am interested in the growth of
$$f_\alpha(n)=\left(\Gamma\left(1+\frac{\alpha}{\sqrt{n}}\right)\right)^n,$$
for positive $\alpha$ and large $n$.
(Preferably something more precise than $1! \leq f_\...
3
votes
1answer
383 views
Integral involving the gamma function
If we define $$f(x) = 1 + \frac{\cos\big(\pi\frac{\Gamma(x) + 1}{x}\big)}{2 - \cos(2\pi{x})}$$ how would one go about evaluating
$$ \int_1^R \frac{1}{x} \log{f(xe^{i\alpha})} dx$$ for some parameter ...
11
votes
3answers
572 views
Complex proof of $B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$
It is a question in spirit of this one.
Is there a way to prove Euler's formula
$$
\int_0^1 x^{a-1}(1-x)^{b-1}dx=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}
$$
using contour integration (and maybe ...
7
votes
2answers
720 views
Calculation of integral using Gamma function when the imaginary part is zero
Consider the following expression of Gamma function
$$\frac{\Gamma(z)}{p^z}=\int_{0}^{\infty}e^{-pt}t^{z-1}dt \ \ \ \ \ \ \ \ \ (1)$$
where $Re(z)>0$ and $Re(p)>0$.
In Lebedevs book "special ...
11
votes
1answer
220 views
Probability distribution derived from gamma function - does it have a name?
Consider the complex gamma function, denoted by $\Gamma(\sigma+it)$.
Now, let's fix $\sigma$ and let t vary. Then consider the following expression:
$$|\Gamma(\sigma+it)|^2$$
For any choice of $\...
14
votes
2answers
624 views
A conjecture about $\lfloor n!\cdot q/e\rfloor-\,!n\cdot q$
I was thinking about this question asked at Math.SE, when I came up with the following conjecture.
For every $q\in\mathbb Q$ consider a sequence $s_n^{(q)}$ (terms within the sequence are indexed by $...
5
votes
1answer
399 views
On a Sum of Gamma Functions
I am working on a problem where the following sum appears:
$$F(s, t)=\frac{1}{\Gamma(1+2\alpha)}\sum_{n=0}^{\infty}{\frac{s^{n} t^{n}}{\left[(s+1)(t+1)\right]^{n+1+\alpha}}\frac{\Gamma(n+1+2\alpha)}{\...
2
votes
0answers
183 views
Continued Fraction: Please prove $\frac{1}{e \gamma (x+1,1)}=x+\frac{1}{x+1+\frac{2}{x+2+\frac{3}{x+3+\frac{4}{\dots}}}}$ [closed]
I have been playing around with Mathematica and continued fractions and I noticed something.
ContinuedFractionK[n, n + x, {n, 1, Infinity}] ==-x + 1/(E Gamma[1 + x] - E Gamma[1 + x, 1])==-x + 1/(E ...
1
vote
0answers
522 views
Proving injectivity of a multivariable function
Let $I$ denote the interval $(0,\infty)$, we define the function $f:I^2\to I^2$ by,
$$f(x,y)=\left({\Gamma(4x+y)\Gamma(y)\over {\Gamma(2x+y)}^2},{\Gamma(4x+y)\Gamma(2x+y)\over {\Gamma(3x+y)}^2}\right)$...
7
votes
0answers
417 views
What is the $q$-analog of $\Gamma(z)\Gamma(1-z)=\frac\pi{\sin(\pi z)}$?
I would expect the $q$-Gamma function to have the property which would be the $q$-analog of the Euler reflection formula from my question title.
More concretely: $\Gamma(z)$ has simple poles at ...
4
votes
2answers
526 views
Do the roots of this equation involving two Euler products all reside on the critical line?
This question loosely builds on the second part of this one.
Take the Riemann $\xi$-function: $\xi(s) =\frac12 s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s)$. Numerical ...
2
votes
0answers
139 views
Are all complex zeros of $\Psi^{(s)}(1) \pm \Psi^{(1-s)}(1)$ on the critical line?
The balanced polygamma function $\Psi^{(s)}(x)$ for $x=1$ can be expressed as:
$$\Psi^{(s)}(1)=\dfrac{\big(\Psi(-s)+\gamma\big)\,\zeta(s+1)+\zeta'(s+1)}{\Gamma(-s)}$$
Note $\Psi(s)$ is the digamma ...
4
votes
0answers
97 views
how understand periodicity in a combination of power, gamma and zeta functions?
Riemann's functional equation may be written:
$$
\frac{\zeta(s)}{\zeta(1-s)} = 2^s \pi^{s-1} \sin(\frac{\pi s}2) \Gamma(1-s) \tag{1}
$$
and so by symmetry:
$$
\frac{\zeta(1-s)}{\zeta(s)} = 2^{1-s} \pi^...
6
votes
1answer
187 views
An identity of complicated combinations of gamma functions (related to hypergeometric functions)
Can somebody help me in proving the following equation?
\begin{align*}&\textstyle \sum _{d=0} ^{n} \frac{1}{d!(n-d)!} \frac{\Gamma (b+d) \Gamma (b+n-d) \Gamma (c-n+d) \Gamma (c-b+1-n + 2d) \...
9
votes
0answers
407 views
Do all complex zeros in the strip of $\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$ lie on the critical line?
Numerical evidence suggests that the complex zeros of:
$$f(s):=\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$$
all reside on the line $\Re(s)=\frac12$, except for a finite few outside ...
5
votes
0answers
314 views
A relation between the Gamma function and the Mobius function?
It is well known how altering the integral for the Gamma function:
$$\displaystyle \Gamma(s) = \int_0^\infty t^{s-1} e^{-t}\,dt$$
through substituting $t=nx$,
$$\displaystyle \Gamma(s)\frac{1}{n^s} ...
8
votes
1answer
2k views
Algorithm to produce random number with a gamma distribution
I'd like to produce pseudo-random numbers with different distributions for a Monte Carlo simulation.
I've got the poisson distribution working nicely with an algorithm from Knuth. I'm having trouble ...
