Questions tagged [gamma-function]
used only for functions based on gamma, not functions with some obscure relation to gamma
167
questions
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Why is the Gamma function shifted from the factorial by 1?
I've asked this question in every math class where the teacher has introduced the Gamma function, and never gotten a satisfactory answer. Not only does it seem more natural to extend the factorial ...
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votes
8
answers
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Inverse gamma function?
This is an analysis question I remember thinking about in high school. Reading some of the other topics here reminded me of this, and I'd like to hear other people's solutions to this.
We have the ...
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votes
6
answers
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Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real part = $\frac12$ ?
The function $\Gamma(s)$ does not have zeros, but $\Gamma(s)\pm \Gamma(1-s)$ does.
Ignoring the real solutions for now and assuming $s \in \mathbb{C}$ then:
$\Gamma(s)-\Gamma(1-s)$ yields zeros at:
...
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24
votes
2
answers
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Importance of Log Convexity of the Gamma Function
The Bohr-Mollerup theorem states that the Gamma function is the unique function that satisfies:
1) f(x+1) = x*f(x)
2) f(1) = 1
3) ln(f(x)) is convex
The Gamma function is meant to interpolate the ...
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votes
4
answers
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views
Who invented the gamma function?
Who was the first person who solved the problem of extending the factorial to non-integer arguments?
Detlef Gronau writes [1]: "As a matter of fact, it was Daniel Bernoulli who gave in 1729 the ...
- 1,054
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votes
3
answers
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views
Is this combination of generalized polygamma and dilogarithm actually zero? $\Im\;\psi^{(-2)}(1+i)+\frac1{4\pi}\text{Li}_2(e^{-2\pi})-\log\sqrt{2\pi}+\frac{5\pi}{24}+\frac12$
I encountered this quantity in my calculations and tried to simplify it. Approximate numeric calculations suggested it could be zero (more precisely, it is certainly less than $10^{-4\times10^3}$ in ...
- 1,679
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2
answers
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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 ...
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2
answers
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Why does the Gamma function satisfy a functional equation?
In question #7656, Peter Arndt asked why the Gamma function completes the Riemann zeta function in the sense that it makes the functional equation easy to write down. Several of the answers were from ...
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votes
2
answers
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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 ...
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2
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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 $...
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answer
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Transcendence of $\Gamma(1/3), \Gamma(1/4)$
This is a re-post from MSE as I did not get even a single comment there.
Wikipedia mentions that the transcendence of $\Gamma(1/3), \Gamma(1/4)$ was proved by G. V. Chudnovsky. Does anyone have a ...
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13
votes
1
answer
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Can a harmonic number be a rational number for non-integer rational argument?
Define harmonic numbers for a complex argument $z$ as $H_z=\frac{\Gamma'(z+1)}{\Gamma(z+1)}-\Gamma'(1)$.
For $n\in\mathbb{N}$, $H_n$ are usual harmonic numbers $\sum^n_{k=1} k^{-1}$ . They are ...
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votes
6
answers
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One-line proof of the Euler's reflection formula
A popular method of proving the formula is to use the infinite product representation of the gamma function. See ProofWiki for example.
However, I'm interested in down-to-earth proof; e.g. using the ...
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$\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$
I'm trying to solve the integral
$\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$,
where $s$, $r$ and $m$>1 are positive integers.
My question is whether a closed form ...
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votes
1
answer
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New method to compute square roots [closed]
In 2011 when I was in school I created a formula to calculate square roots... For $x\in\mathbb{R}$ with $x>0$ the following holds:
$$\sqrt{x} = \sum_{n=0}^{\infty}\frac{\left(\prod_{k=1}^{n}\left(\...
- 53
11
votes
1
answer
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Probability distribution derived from gamma function - does it have a name?
Consider the complex gamma function, denoted by $\Gamma(\sigma+it)$.
Now, let's fix $\sigma$ and let t vary. Then consider the following expression:
$$|\Gamma(\sigma+it)|^2$$
For any choice of $\...
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11
votes
3
answers
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Complex proof of $B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$
It is a question in spirit of this one.
Is there a way to prove Euler's formula
$$
\int_0^1 x^{a-1}(1-x)^{b-1}dx=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}
$$
using contour integration (and maybe ...
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votes
0
answers
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Are the twin primes the only positive double zeros of this real function?
Agno's answer
was extremely helpful.
For $x \in \mathbb{R}, x \ge 1$ define
$$ f(x) = \sin\frac{\pi(\Gamma(x)+1)}{\lfloor x \rfloor}$$
By Wilson's theorem the positive integer zeros of $f(x)$ are ...
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answer
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"unexpected" residue formula for $\Gamma^3(s)/(\Gamma(3s)(e^{2\pi is}-1)) $
There is a related problem in my current work: to find the residue of the following function at any negative integer $s=-n$:
$$f(s)=\frac{\Gamma^3(s)}{\Gamma(3s)(e^{2\pi is}-1)}$$
It seems to be a ...
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answer
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Errata for Emil Artin's 'The Gamma Function'?
In the English translation of The Gamma Function by Emil Artin (1964 - Holt, Rinehart and Winston) there appears to be a mistake in the formula given for the gamma function on page 24:
$$\Gamma(x) = \...
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votes
3
answers
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Inequality for a gamma function
Let $s=\sigma+it$ and $\Gamma(s)$ be the Euler gamma function.
Does the inequality hold?
$$
\left|\frac{\Gamma(s)}{\Gamma(2-s)}\right|\leq |s|^{2(\sigma-1)},\, 1<\sigma<2,\, t\in \mathbb{R}.
$$...
9
votes
0
answers
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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 ...
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votes
3
answers
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No simple duplication formula for factorials?
Many special functions including the gamma function have a duplication formula of some sorts. In the case of the gamma function it reads:
Gamma(2z) = Gamma(z) Gamma(z+1/2) 22z-1/Gamma(1/2)
On the ...
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3
answers
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Multiplicative integral of $\Gamma(x)$
A recent question on the notion and notation of multiplicative integrals
( What is the standard notation for a multiplicative integral? ) induced me to play with the Riemann products of the Gamma ...
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An interesting infinite product involving the factorial function with connection to the K and gamma function
I have posted this question in StackExchange, but it didn't get any answers there. This question is important for my research. I got stuck on an infinite product which even WolframAlpha can't answer. ...
user167505
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1
answer
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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 ...
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1
answer
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Converse of a result of Koblitz and Ogus on algebraic products of gamma values
Let $n$ be a positive integer, $a_1,\ldots,a_{n-1}\in\mathbb{Z}$. Suppose that for every $u\in(\mathbb{Z}/n\mathbb{Z})^\times$,
$$
\tag{$\star$}
\sum_{i=1}^{n-1} i a_{(ui\!\!\!\mod n)} = 0.
$$
Then ...
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votes
1
answer
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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 ...
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votes
0
answers
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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 ...
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3
answers
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Method to evaluate an infinite sum of ratio of Gamma functions (how does Mathematica do it?)
This question arose from Amdeberhan's question, the evaluation of a double integral, which can be reduced to the evaluation of this series:
$$\sum _{n=0}^{\infty } \frac{\Gamma \left(n+\frac{1}{2}\...
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votes
2
answers
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Reference request: proofs of integrals presented in Erdélyi's *Table of Integral Transforms*
I have asked related questions on MSE and received no answer, so: I am looking for proofs of the integrals presented in Erdélyi's Table of Integral Transforms vol. i-ii, specifically proofs of the ...
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answers
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Calculation of integral using Gamma function when the imaginary part is zero
Consider the following expression of Gamma function
$$\frac{\Gamma(z)}{p^z}=\int_{0}^{\infty}e^{-pt}t^{z-1}dt \ \ \ \ \ \ \ \ \ (1)$$
where $Re(z)>0$ and $Re(p)>0$.
In Lebedevs book "special ...
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votes
1
answer
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Conjectured bound on Kummer's function (confluent hypergeometric function)
I believe the following bound to be correct:
${}_1F_1(-a;1+a;-z) a z^{-a} \gamma(a,z) \leq 1$
for real-valued a > 0 and z $\geq 0$. $\gamma(a,z)$ is the lower incomplete gamma function.
Apart from ...
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4
answers
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Integral transform and $\frac{1}{n!}$.
Probably this is a trivial question, but I am unable to find an answer: is there a function $v(x)$ such that
$$
\int_{0}^\infty x^n e^{v(x)} dx =\frac{1}{n!}
$$
for all positiv integer n?
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1
answer
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Gamma function versions of combinatorial identites?
We can extend the binomial coefficient $\binom{n}{k}$ to $\mathbb{R}$ or $\mathbb{C}$ by defining $\binom{x}{y}=\frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma(x-y+1)}$. Do any the standard binomial coefficient ...
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1
answer
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An identity of complicated combinations of gamma functions (related to hypergeometric functions)
Can somebody help me in proving the following equation?
\begin{align*}&\textstyle \sum _{d=0} ^{n} \frac{1}{d!(n-d)!} \frac{\Gamma (b+d) \Gamma (b+n-d) \Gamma (c-n+d) \Gamma (c-b+1-n + 2d) \...
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0
answers
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Relation between Gross-Koblitz and Chowla-Selberg formulas
The Chowla-Selberg formula relates the eta function with values of the gamma function at rational numbers. The eta function appears, at least in the proofs I have seen, related to values of $L$-...
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3
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Evaluating the series $\sum_{n=0}^{\infty} n! x^n$ and inverse variable-fractional-derivatives
So I was interested in formally assigning values to the completely divergent series $G(x) = \sum_{n=0}^{\infty} n!x^n $. I guess the question COULD end here if you already have an idea of how to ...
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votes
3
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Asymptotic behavior of integral with gamma functions
Consider the following function defined for complex numbers $z\in\mathbb{C}$ with $\Re(z)\geq \frac{1}{2}$:
$$F(z)=\frac{1}{5^{\Re(z)}}\int_0^\infty \left| \frac{\Gamma(z+ix)\Gamma(z-ix)}{\Gamma(z)^2}...
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2
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Extended binomial coefficients and the gamma function
For which $(a,b,n) \in \mathbb{Z}^3$ satisfying $a+b=n$ does $\frac{\Gamma(z+1)}{\Gamma(x+1)\Gamma(y+1)}$ approach a limit as $(x,y,z) \rightarrow (a,b,n)$ in $\mathbb{C}^3$, and what is that limit? (...
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On a Sum of Gamma Functions
I am working on a problem where the following sum appears:
$$F(s, t)=\frac{1}{\Gamma(1+2\alpha)}\sum_{n=0}^{\infty}{\frac{s^{n} t^{n}}{\left[(s+1)(t+1)\right]^{n+1+\alpha}}\frac{\Gamma(n+1+2\alpha)}{\...
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1
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How to evaluate inverse Laplace transform of $e^{- \sqrt{s}} $ using series?
I tried to find an inverse Laplace transform by series as follows
$$ f(t)=L^{-1}_s\left(e^{-\sqrt{s}}\right)(t)=L^{-1}_s\left(\sum_{k=0}^{\infty}\frac{(-1)^k}{k!} s^{\frac{k}{2}}\right)(t)$$
and by ...
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votes
1
answer
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On the integral $\int_0^1\log(x!)dx$ revisited
I was interested in an integral that I known from [1], it is
$$\int_0^1 \log(x!)dx.$$
I tried to get such closed-form using myself ideas and symbolic calculations, also with the help of Wolfram ...
5
votes
1
answer
387
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Computing the $p$-adic gamma function $\Gamma_p$
Let $p>2$ be a prime. For $n \in \mathbb{Z}^+$ we can define
\begin{equation}
F(n) = (-1)^n \prod_{1<i<n, p\nmid i} i.
\end{equation}
Since $\mathbb{Z}$ is dense in $\mathbb{Z}_p$, we can ...
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votes
2
answers
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views
Is $\Gamma(s, x=s-1)/\Gamma(s)$ decreasing for real $s>1$? Is $\Gamma(s, x=s)/\Gamma(s)$ increasing?
This has received no full solution at StackExchange.
As per https://dlmf.nist.gov/8.10#E13 we have
$$\frac{\Gamma\left(n,n\right)}{\Gamma\left(n\right)}<\frac{1}{2}<\frac{\Gamma%
\left(n,n-1\...
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votes
1
answer
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How naturally can functions defined by parametric integrals be interpolated from $\mathbb N$ to $\mathbb R^+$?
It is well known that several definite integrals $I_n$ containing a parameter $n\in\mathbb N$ can be expressed recursively (e.g. doing integration by parts) in terms of $I_{n-1} $ or $I_{n-2} $, and ...
- 13.2k
5
votes
1
answer
632
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
$$\psi(x)=\...
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votes
2
answers
246
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Inverse Mellin transform of 3 gamma functions product
I want to calculate the inverse Mellin transform of products of 3 gamma functions.
$$F\left ( s \right )=\frac{1}{2i\pi}\int \Gamma(s)\Gamma (2s+a)\Gamma( 2s+b)x^{-s}ds$$
Above contour integral has ...
- 59
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votes
3
answers
346
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The exact constant in a bound on ratios of Gamma functions
The answer to another question (Upper bound of the fraction of Gamma functions) gave an asymptotic upper bound for an expression with Gamma functions:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\...
- 343
5
votes
1
answer
466
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On the multidimensional generalisation of Gamma function
Gamma function is defined as
$$
\Gamma(z) = \int\limits_{0}^{+\infty} x^{z-1} e^{-x} \; dx
$$
I'm looking for multidimensional generalisation of this definition. I consider the class $Q$ of positive,...
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