# Questions tagged [gamma-function]

used only for functions based on gamma, not functions with some obscure relation to gamma

119 questions
Filter by
Sorted by
Tagged with
39 views

### How can I show that this product is equal to a product of Gamma functions? [migrated]

$\prod_{n=0}^{x-1}\left( 1+\frac{a}{an+b}\right) = \frac{\Gamma\left(\frac{a}{b}\right)\Gamma\left(x+\frac{a+b}{b}\right)}{\Gamma\left(\frac{a+b}{b}\right)\Gamma\left(x+\frac{a}{b}\right)}$ I found ...
146 views

### Any simplification of this inequality if it is true? :For $t\geq 1.22$: $|\zeta(0.5+it)|\leq 0.5 \frac{|\Gamma(0.5+it)|}{|\Gamma(-0.5+it)|}$

When I tried to give bounds for $\zeta(0.5+it)$ using some transformations over Gamma function using the function $f(x)=\exp(-n x)$ over the range $(0,+\infty)$ , For $Re(s)=\frac12$ and $t >0$ ...
531 views

### Reference request: proofs of integrals presented in Erdélyi's *Table of Integral Transforms*

I have asked related questions on MSE and received no answer, so: I am looking for proofs of the integrals presented in Erdélyi's Table of Integral Transforms vol. i-ii, specifically proofs of the ...
584 views

### Formula for volume of $n$-ball for negative $n$

Does the expression $$\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}R^n,$$ which gives the volume of an $n$-dimensional ball of radius $R$ when $n$ is a nonnegative integer, have any known ...
33 views

### Bounding the working precision required in Spouge's Approximation

Spouge's approximation for the gamma function is $\Gamma(z+1) = (z+a)^{z+\frac{1}{2}}e^{-z-a} \left(c_0 + \sum_{k=1}^{a-1} \frac{c_k}{z+k} + \epsilon_a(z) \right)$ where the coefficients are given ...
92 views

150 views

124 views

### An equation with Gamma Euler function in critical strip

Let $$D=\{z \in \mathbb{C} : 0<\Re(z)<\frac{1}{2} \text{ or } \frac{1}{2}<\Re(z)<1 \}$$ that is the critical strip without critical line. I have to find if the following equation, with ...
191 views

323 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 ...
111 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 ...
187 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-...
142 views

435 views

351 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. ...
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 $\sigma,x \in \mathbb{R}$: $$\frac{1}... 1answer 144 views ### Closed form for sum involving digamma? [closed] Let \Gamma(n) be Euler's Gamma function and \psi_0 = \frac{\Gamma'(n)}{\Gamma(n)} be the Digamma function. Is there a closed form for$$\sum_{n=1}^{\infty} \frac{\psi_0(n)}{n^2}=? I've done ...
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 ...