Questions tagged [gamma-function]
used only for functions based on gamma, not functions with some obscure relation to gamma
158
questions
11
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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:
$$\sqrt{x} = \sum_{n=0}^{\infty}\frac{\left(\prod_{k=1}^{n}\left(\...
- 45
2
votes
2
answers
468
views
Integral calculus with Gamma function [closed]
I have to prove that for $0<\alpha<1$ and $\beta>0$,
\begin{equation}
\int_{0}^{\infty} x^{-\alpha-1}\left(e^{-\beta x}-1\right)dx=\beta^\alpha\Gamma(-\alpha),
\end{equation}
and I have ...
- 57
0
votes
1
answer
133
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Infinite limit of sums of gamma functions is constant?
The following expression arises in the study of hierarchical models. I suspect that the sum of the underlined $4$ terms become constant as $\alpha\rightarrow \infty$. Mathematica agrees when prompted ...
- 111
3
votes
0
answers
180
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Derivation of an integral containing the complete elliptic integral of the first kind
I found the following formula in "INTEGRALS AND SERIES, vol.3" by Prudnikov, Brychkov and Marichev (page 188, eq.5).
$$\int_0^{\infty} \frac{x^{\alpha-1}}{\sqrt{(a+x)^2+z^2}}K(\frac{2\sqrt{...
- 31
3
votes
4
answers
486
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Some Log integrals related to Gamma value
Two years ago I evaluated some integrals related to $\Gamma(1/4)$.
First example:
$$(1)\hspace{.2cm}\int_{0}^{1}\frac{\sqrt{x}\log{(1+\sqrt{1+x})}}{\sqrt{1-x^2}} dx=\pi-\frac{\sqrt {2}\pi^{5/2}+4\sqrt{...
user497425
0
votes
0
answers
48
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Incomplete Gamma function $\Gamma(0,x)$ and $\Gamma(0,-x)$
I want to find the value of this
\begin{align}
y=\Gamma(0,x)-\Gamma(0,-x)
\end{align}
where $\Gamma$ is the upper incomplete Gamma function, $x>0$ is real. I can't find the definition of $\Gamma(0,-...
- 11
1
vote
0
answers
64
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polynomial approximation of hypergeometric function 2F1
I have the following function $T(k_1,k_2)$ resulting from multiphoton transition matrix elements calculations:
$T(k_1,k_2)=\gamma^{-k_2}\sum_{j=0}^{k_1}(j+2)_{l+1}\binom{k_1}{j}(k_1+1)_3(\gamma-1)^{j}{...
- 11
0
votes
1
answer
158
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Conjecture about the equality : $f(y)=y\ln(y)+\sum_{n=2}^{\infty}\pm\frac{\left(y\ln y\right)^n}{2^{a_n}}$
I try here because I expect I cannot have any answer on MSE :
Problem :
Let :
$$f\left(x\right)=\frac{\left(1+\ln27\right)x!\ln x!}{x+1},g\left(x\right)=x\ln x$$
Then it seems $\exists y\in(0,1)$ and $...
- 97
5
votes
3
answers
254
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Evaluating the series $\sum_{n=0}^{\infty} n! x^n$ and inverse variable-fractional-derivatives
So I was interested in formally assigning values to the completely divergent series $G(x) = \sum_{n=0}^{\infty} n!x^n $. I guess the question COULD end here if you already have an idea of how to ...
- 1,450
0
votes
0
answers
74
views
An interpolation of $n!$ such that its derivatives have few zeros
The $\Gamma$-function restricted to $(0,+\infty)$ has the following properties:
$\Gamma(n)=(n-1)!$ for $n=1,2,3,...$.
The $k$'th derivative $\Gamma^{(k)}$ has no zeros on $(0,+\infty)$ when $k$ is ...
- 475
5
votes
0
answers
390
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Determinant of Hankel matrix with $a_n=(n!)^2$
Consider a Hankel matrix of the form
$H_n(a_0(n))=\begin{pmatrix}
a_0(n) & (1!)^2 & (2!)^2 & \cdots & (n!)^2\\
(1!)^2 & (2!)^2 & (3!)^2& \cdots & ((n+1)!)^2\\
(2!)^2 &...
- 51
1
vote
0
answers
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Uniform bound on product of Gamma functions in an article by Jerison and Kenig
I am new here. I'm reposting a question I originally posted here on Math Stack Exchange. I realized that maybe this is more appropriate place to ask such a question...
I have been trying to read ...
- 11
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votes
0
answers
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Does it make sense to express upper bounds on arithmetic sequences with Dirichlet generating functions?
In order to see what happens when taking the functional equation in this form:
$$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s) $$
$$\xi(s) = \xi(1 - s)$$
$$\pi^{-s/2}\ \Gamma\left(\...
- 1,053
5
votes
2
answers
190
views
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
1
vote
2
answers
331
views
Power series of ratio of Gamma functions
Let $a>1$ and define $G_a(x)=\sum\limits_{n=0}^{+\infty} \frac{\Gamma(\frac{2n+1}{a})}{\Gamma(2n+1)\Gamma(\frac{1}{a})}x^n$ where $\Gamma$ is the Gamma function. This series is convergent on $\...
- 29
2
votes
0
answers
98
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Motivation behind the Bohr-Mollerup Theorem relating the Gamma function
In Wikipedia, it states about the Bohr-Mollerup Theorem:
The theorem was first published in a textbook on complex analysis, as Bohr and Mollerup thought it had already been proved.
If anyone knows, ...
- 143
1
vote
1
answer
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Stirling's formula and a Gamma function relation
I am trying to understand a paper by by A. Booker on poles of Artin $L$-functions where in one of the lemmas he uses the following identity, derived using Stirling's formula:
$$
\frac{\Gamma(s/2)^2}{2^...
- 31
5
votes
2
answers
275
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Extended binomial coefficients and the gamma function
For which $(a,b,n) \in \mathbb{Z}^3$ satisfying $a+b=n$ does $\frac{\Gamma(z+1)}{\Gamma(x+1)\Gamma(y+1)}$ approach a limit as $(x,y,z) \rightarrow (a,b,n)$ in $\mathbb{C}^3$, and what is that limit? (...
- 18.8k
0
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0
answers
91
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Addition formulas for q-analogs of trigonometric functions
Sine and Cosine functions possess notable formulas for addition of angles
$$
\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b) \qquad \text{or} \qquad \cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b).
$$
One can ...
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2
votes
1
answer
150
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Gamma function and the somewhat extended version of Bohr-Mollerup theorem
The Gamma function $\Gamma$ is defined by
\begin{equation*}
\Gamma(x)=\int_{0}^\infty t^{x-1}e^{-t} \,\mathrm{d}t,
\end{equation*}
for $x>0$. It satisfies the well-known functional equation
$$\...
- 143
5
votes
0
answers
639
views
Nature of function as $x\rightarrow\infty$
I'm studying the limits and applicability of Abel Plana summation for different test functions (class of functions). In doing so this just pops out and couldn't handle the said integral so asked here (...
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1
vote
1
answer
85
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Simplification of $\sum_{m=0}^\infty \text B_z(m+a,b-m)x^m,\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}$ in terms of Kampé de Fériet function
Here is the goal sum where the Pochhammer Symbol with the Incomplete Beta function series
$$\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}=\sum_{m=0}^\infty z^{m+a}\sum_{n=0}^\infty\frac{(1-(b-m))...
- 125
5
votes
0
answers
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The Basel problem revisited?
In the Basel problem, the $sinc$ function is considered at the Wikipedia page.
Let me try to make an alternative function definition:
$$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \...
- 1,819
3
votes
2
answers
246
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$\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} z^{2k}$ is an elementary function
I try to calculate the following series
\begin{align*}
S_{n,m}(z)=\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} \, z^{2k},
\end{...
- 630
2
votes
1
answer
232
views
Upper bound for the complex Beta function
The question is almost the same as here.
What is the upper bound for a complex Beta function $\displaystyle B(s,z)=\frac{\Gamma(s) \Gamma(z)}{\Gamma(s+z)}$ with $0<Re(s)<1$ and $0<Re(z)<1$;...
- 29
1
vote
1
answer
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Sum of reciprocal of Pochhamer symbols through multiples of a natural L
In a question in StackExchange (https://math.stackexchange.com/questions/4236635/sum-of-quotients-of-gamma-functions), I asked if there is a closed expression for the following sum related with a ...
1
vote
1
answer
234
views
How to prove the convexity of a simple function involving a ratio of two polygamma functions?
Let
\begin{equation*}
\Gamma(z)=\int_0^{\infty}t^{z-1}\textrm{e}^{-t}\textrm{d}t, \quad \Re(z)>0
\end{equation*}
and
$$
\psi(z)=[\ln\Gamma(z)]'=\frac{\Gamma'(z)}{\Gamma(z)}.
$$
In the literature, ...
- 706
4
votes
1
answer
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views
Beta function, harmonic numbers, and integral values
A problem of Mathematical Physics that I am working on involves the computation of a certain integral. Part of the result reads:
$$
I_k:= [\beta_x( -1 - k, 0) + H_{-2 - k}]x^k
$$
where $\beta_x( -1 - ...
- 225
4
votes
2
answers
165
views
Hankel determinant of incomplete gamma functions
I have some expressions that involve Hankel determinants of incomplete gamma functions. They are of the ($r \times r$ form)
I'd like to evaluate these determinants. Elementary operations help, but ...
- 41
3
votes
0
answers
190
views
definite integral with incomplete gamma function and exponential
While working with electron density computations in quantum chemistry, I encountered the following improper integral:
$$
I(k, n) = \int\limits_0^\infty \Gamma\left(\frac{3}{n},\ k r^n\right) r \exp(-k ...
- 31
7
votes
3
answers
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views
Method to evaluate an infinite sum of ratio of Gamma functions (how does Mathematica do it?)
This question arose from Amdeberhan's question, the evaluation of a double integral, which can be reduced to the evaluation of this series:
$$\sum _{n=0}^{\infty } \frac{\Gamma \left(n+\frac{1}{2}\...
- 163k
5
votes
1
answer
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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 ...
- 649
2
votes
0
answers
53
views
Class of differentiated Gamma functions: are there any algebras where they are elementary?
There is a class of differentiated gamma functions (DGFs) that basically can be expressed via derivatives of Gamma function.
They include the Gamma function, Polygamma function, and Hurwitz Zeta ...
- 8,838
2
votes
3
answers
430
views
how to numerically evaluate $\int_{0}^{\infty} \frac{1}{x!} dx$ [closed]
So I was graphing the equation $ y=\frac{1}{x!} $ for $ x \geq 0$ and tried the integral:
$$\int_{0}^{\infty} \frac{1}{x!} dx$$
$$\int_{0}^{\infty} \frac{1}{\Gamma(x+1)} dx$$
$$\int_{0}^{\infty} \frac{...
- 57
2
votes
1
answer
256
views
Analytic continuation of convergent integral
I was trying to solve the following integral:
$$I = \oint _{|z|=1}\frac{dz}{2 \pi i z}\int_{0}^{\infty} dr \dfrac{e^{-\tfrac{r^2}{z^2}}r^{2n+1}}{z^2(z-1)} $$
The singular structure in the $z$ ...
- 121
8
votes
2
answers
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views
An interesting infinite product involving the factorial function with connection to the K and gamma function
I have posted this question in StackExchange, but it didn't get any answers there. This question is important for my research. I got stuck on an infinite product which even WolframAlpha can't answer. ...
user167505
5
votes
2
answers
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views
Is $\Gamma(s, x=s-1)/\Gamma(s)$ decreasing for real $s>1$? Is $\Gamma(s, x=s)/\Gamma(s)$ increasing?
This has received no full solution at StackExchange.
As per https://dlmf.nist.gov/8.10#E13 we have
$$\frac{\Gamma\left(n,n\right)}{\Gamma\left(n\right)}<\frac{1}{2}<\frac{\Gamma%
\left(n,n-1\...
- 794
4
votes
0
answers
157
views
Inverse mellin transform
Let $K_1(t)$ be the K-Bessel function, then we have
$\int_{0}^{\infty}K_v(y)y^s\frac{dy}{y}=2^{s-2}\Gamma(\frac{s+v}{2})\Gamma(\frac{s-v}{2})$ See page 106 of Bump's book Automorphic forms and ...
- 609
0
votes
0
answers
103
views
What is the value of this product (with gamma and zeta function)?
Since $\zeta(s)=0$ and $\Gamma(s)$ has poles at even negative numbers $-2, -4, -6...$ Can we find a value for $\frac{1}{\zeta(s)\Gamma(s)}$ using L'Hôpital's rule?
- 150
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votes
1
answer
172
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 $t >0$ ...
- 2,198
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votes
2
answers
610
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 ...
16
votes
2
answers
1k
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 ...
- 18.8k
4
votes
3
answers
306
views
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:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\...
- 309
1
vote
1
answer
141
views
Inequalities involving Gamma function
I am having some difficulty in proving the following inequality:
\begin{equation*}
\frac{1-e^{-\gamma b}}{b^\eta}-\frac{1-e^{-\gamma s}}{s^\eta}\geq \gamma(1-\eta)\int^b_sy^{-\eta}e^{-\gamma y}dy
\end{...
- 33
1
vote
2
answers
309
views
Closed form of $\prod_{i=0}^{N}\big(i!\big)^{{N}\choose{i}}$
I have made a question here about closed form of the following:
$$\prod_{k=0}^{N}\big(k!\big)^{{N}\choose{k}}$$
I know that there is a known closed form for,
$$\prod_{i=0}^{N}\big(i!\big)=G(N+2)$$
...
- 155
0
votes
2
answers
203
views
Interpolating asymptotic expression for logarithm of middle binomial sums
Define $S(k,2n)=\sum_{i=-k}^k\binom{2n}{n+i}$ at every $k\in\{0,\dots,n\}$.
We know $$\ln(S(\gamma\mbox{ } n^\gamma,2n))\asymp(2n\ln2-\frac12\ln\pi-\frac12\ln n)$$ at $\gamma\rightarrow0$ and $$\ln(S(...
- 1,776
0
votes
1
answer
229
views
Logarithm of an integral involving generalized real binomial coefficients
I could not find a closed form for this integral although I think it should have been studied.
What is a good approximation to $I$ in
$$I=\ln\Bigg(\int_{0}^y\binom{2m}{m(1+x)}dx\Bigg)$$ where $0\...
- 1,776
0
votes
1
answer
181
views
The distribution of the power of the sum of inner products of two independent complex normal vectors
If I have $\mathbf x_n=[x_0, x_1,... ,x_K]^T$ and $\mathbf y_n=[y_0, y_2, ..., y_K]^T$, where $x,y\sim\mathcal C\mathcal N(\mathbf 0,\sigma^2\mathbf I)$.
What is the distribution of the following ...
- 105
0
votes
1
answer
535
views
The distribution of the sum of inner products of two independent complex normal vectors
If I have $\mathbf x_n=[x_0, x_1,... ,x_K]^T$ and $\mathbf y_n=[y_0, y_2, ..., y_K]^T$, where $x,y\sim\mathcal C\mathcal N(\mathbf 0,\sigma^2\mathbf I)$.
What is the distribution of the following ...
- 105
1
vote
1
answer
182
views
Asympotics of $\int_0^{\infty} \left[ Q(m,x)\right]^d dx$
I'm interested in the asympotic behaviour (assuming an exact solution is intractable) of $$I(m,d)=\int_0^{\infty} \left[ Q(m,x)\right]^d dx$$
for fixed $d \in \mathbb{N}$ (in particular, for $d=3$) ...
- 298