Questions tagged [gamma-function]
used only for functions based on gamma, not functions with some obscure relation to gamma
182 questions
2 votes
1 answer
262 views
Identities for hypergeometric functions
In my work I came across the hypergeometric function $_3F_2(a,b,c;a-b+3,a-c+3;1)$. Since I need to study the poles of this function, I would prefer to express it in terms of finite ratios of gamma ...
- 3,935
2 votes
1 answer
503 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 $$\DeclareMathOperator{\Im}{Im}\DeclareMathOperator{\Re}{Re} \displaystyle B(s,z)=\frac{\Gamma(s) \Gamma(z)}...
CommunityBot
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1 vote
1 answer
461 views
How to prove the convexity of a simple function involving a ratio of two polygamma functions?
Let \begin{equation*} \Gamma(z)=\int_0^{\infty}t^{z-1}\textrm{e}^{-t}\textrm{d}t, \quad \Re(z)>0 \end{equation*} and $$ \psi(z)=[\ln\Gamma(z)]'=\frac{\Gamma'(z)}{\Gamma(z)}. $$ In the literature, ...
CommunityBot
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0 votes
0 answers
63 views
References on beta function approximation
In this Wikipedia article: we have the following about the beta function approximation: Stirling's approximation gives the asymptotic formula: $$ B(x, y) \sim \sqrt{2\pi} \cdot \frac{x^{x - \frac{1}{2}...
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0 votes
0 answers
130 views
Asymptotic behaviour of the analytic integral
Given $m \ge 0$ and $n$ odd, $ s >0 $ is there a way, for computing asymptotic expansion of following integral: $$ \int_{\mathcal{C}} \int_{\mathcal{C}} \frac{ \Gamma(m + s_1) \Gamma(m + s_2) \...
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0 votes
0 answers
216 views
How to prove the convergence of the following series involving Gamma function?
Consider the following result（$d$ denotes the dimensions and $0<t<T$） $$c\left(\sum_{j=0}^\infty\frac{\Gamma^j(1-\kappa)}{\Gamma((j+1)(1-\kappa))}t^{j(1-\kappa)-\kappa}\right)^{\frac{1}{2}}\leq ...
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1 vote
1 answer
284 views
Prove that the regularized incomplete beta function monotone with each of its parameter
Consider the regularized incomplete beta function $I_x(a, b)$ with $x \in [0,1]$ and $a, b > 0$. I am hypothesizing that the function is monotone decreasing with respect to $a$ and monotone ...
CommunityBot
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6 votes
0 answers
500 views
Is there any known $x\in (0,1) \setminus \left\{\frac 1 2\right\}$? such that a simple closed form for $\Gamma(x)$ exists?
Motivation: A friend of mine was working on a problem and tried to compute $ \Gamma\left(\frac{1}{4}\right) $, thinking it is required to find an exact closed form. I quickly told him that it wasn’t ...
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1 vote
0 answers
164 views
Algebraic relations for $\Gamma$ function
Let $N$ and $n$ be positive integers with $\mathrm{GCD}(n,N)\ne1$. I want to prove the following claim: $\Gamma\left(\frac nN\right)$, $\pi$ and the $\Gamma\left(\frac uN\right)$ ($u\in[1,N-1]$, $\...
- 4,286
2 votes
1 answer
416 views
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: $$ \frac{\Gamma(s/2)^2}{2^...
CommunityBot
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9 votes
1 answer
528 views
A hypergeometric series for $\Gamma(1/4)^4/\pi^3$
Sorry if this comes out of the blue. Looking at old notes of mine, I found the identity $$\dfrac{\Gamma(1/4)^4}{\pi^3}=4+\sum_{n\ge0}\binom{2n+1}{n}^3\dfrac{1}{2^{6n+1}}\;.$$ I cannot remember how I ...
- 1,987
7 votes
2 answers
368 views
An identity of complicated combinations of gamma functions (related to hypergeometric functions)
Can somebody help me in proving the following equation? \begin{align*}&\textstyle \sum _{d=0} ^{n} \frac{1}{d!(n-d)!} \frac{\Gamma (b+d) \Gamma (b+n-d) \Gamma (c-n+d) \Gamma (c-b+1-n + 2d) \...
- 3,935
2 votes
0 answers
124 views
Poisson process subordinated by a gamma process
I am working on a problem and I encountered the following situation: $(N(t): t \ge 0)$ is a Poisson process with parameter $\lambda t $. If $T_{n} = \sum_{i=1}^n W_i$ represents the $n^\text{th}$ ...
- 12.8k
2 votes
1 answer
240 views
Reference for Mellin inversion; Meijer G-function
We have $$\frac {\Gamma (a)}{2^a}=\int _{(c)}\Gamma (s)\Gamma (a-s)\,ds,$$ see e.g. Exercise C.23 of Montgomery and Vaughan's "Multiplicative Number Theory". I would like a similar formula ...
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1 vote
3 answers
300 views
The approximated function of $\mathbb{E}\left\{ \ln\left(1+X\right)\right\}$, where $X\sim\operatorname{Gamma}\left(\kappa\ge1,\theta>0\right)$
Given $X \sim \operatorname{Gamma}(\kappa, \theta)$ with CDF $F_X(\kappa, \theta)$, where $\kappa \geq 1$ and $\theta > 0$, the expected value of $\mathbb{E} \left\{ \ln(1+X) \right\}$ is ...
- 14k
0 votes
0 answers
167 views
how to use Gauss Multiplication Formula for Gamma function?
I asked this question in MSE I studied Gauss Multiplication Formula which known for $n\in Z^+ \wedge nx\notin Z^-\cup\{0\}$ $$\Gamma(nz)=(2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\Gamma\left(z+\...
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2 votes
0 answers
72 views
Good Polynomial lower estimates for beta function
I'm looking for polynomial lower estimates for beta function, and what I've found so far is this, which can be found in proposition 2.3 in this paper Proposition 2.3 1. If $0<𝑞<1$ and $𝑝 \geq ...
- 687
1 vote
0 answers
157 views
Poles/Residues of the Gamma function under action of Mobius transform $\Gamma(A(z))$
I am not sure whether this is rather an MO or MSE question but it results from my research, so I put it here. In my effort to find (or to disprove the existence of) $k,l,h\in\mathbb{N}$ such that $2^{...
- 281
7 votes
1 answer
945 views
Representing $\Gamma(a-x)$ in terms of $\Gamma(kx)$ and $\Gamma(a)$ and elementary functions
I asked this question on MSE here. I wonder if it is possible to represent $\Gamma(a-x)$ in terms of powers of $\Gamma(a)$, powers of $\Gamma(kx)$, and elementary functions. I am not looking for any ...
- 120k
174 votes
11 answers
29k views
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 ...
CommunityBot
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4 votes
2 answers
299 views
Eisenstein $E_2$ at imaginary quadratic arguments
In the paper On Epstein's zeta-function, Chowla and Selberg give a formula for evaluating the Dedekind eta function $$\eta (\tau)=e^{\pi i\tau/12}\prod_{n=1}^\infty (1-e^{2\pi i n\tau}),\quad \Im\tau\...
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3 votes
1 answer
713 views
Separating Gamma in two independent functions
I've encountered a problem in my PhD. I would greatly appreciate any suggestions, tips, or comments you might have. The problem is Let $\Gamma(s,x)$ be the incomplete gamma function. Given integers $n ...
- 143k
1 vote
0 answers
60 views
Exponential-like function equivalent for the Dixonian Elliptics
Is there some exponential-like function that acts as partner for the intricate Dixonian Elliptics, in a similar way that the Exponential function acts as a partner for the trigonometric functions ?
8 votes
1 answer
2k views
The inverse of the digamma function
The gamma function is increases on the interval $(x_0, \infty),$ where $x_0$ denotes the unique zero of the digamma function on the positive half line. The inverse function of gamma function defined ...
- 1,465
5 votes
1 answer
820 views
How to evaluate inverse Laplace transform of $e^{- \sqrt{s}} $ using series?
I tried to find an inverse Laplace transform by series as follows $$ 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)$$ and by ...
- 201k
0 votes
1 answer
188 views
Solution or approximation to $\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx$
I'm looking for a solution or approximation to the following indefinite integral $$\int x^{-a} \Gamma\left( b, c x^{-d} + e \right) dx.$$ I've tried Mathematica, but it does not converge to a solution....
- 1,465
60 votes
8 answers
37k views
Inverse gamma function?
This is an analysis question I remember thinking about in high school. Reading some of the other topics here reminded me of this, and I'd like to hear other people's solutions to this. We have the ...
18 votes
2 answers
1k views
Formula for volume of $n$-ball for negative $n$
Does the expression $$\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}R^n,$$ which gives the volume of an $n$-dimensional ball of radius $R$ when $n$ is a nonnegative integer, have any known ...
- 10.4k
0 votes
1 answer
178 views
Is $\Gamma(z,1)\not=0$ for all $z$ with $\Re(z)<0$?
I found this paper online which appears to present zeros of the incomplete gamma function within the right half plane. It makes me think that there are no zeros in the left half plane. Not sure how to ...
- 115k
1 vote
1 answer
236 views
T functions arising from derivatives of incomplete Gamma function
Here the derivatives of the incomplete gamma functions are described via: $$ T(m,s,x) = G_{m-1,\,m}^{\,m,\,0} \!\left( \left. \begin{matrix} 0, 0, \dots, 0 \\ s-1, -1, \dots, -1 \end{matrix} \; \right|...
- 143k
12 votes
6 answers
20k views
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 ...
- 6,830
1 vote
1 answer
131 views
Supremum or upper bound of bivariate function involving logarithms and combinatorial coefficients or the gamma function over a region of the integers
This is a repost from MSE because I got no answers there. I have been trying to find the supremum of this bivariate function over a specific region. However, the expressions that I get are horrible. I ...
- 143k
1 vote
0 answers
139 views
Question on Artin's Gamma function on $\operatorname{SO}(2,0)(\mathbb R)$
$\DeclareMathOperator\SO{SO}$Let $G=\SO(2,0)(\mathbb{R})$, a quasi-split group with signature $(2,0)$. Let $e$ be an element in $O(2,0)(\mathbb{R}) \setminus \SO(2,0)(\mathbb{R})$. Let $\pi$ be an ...
- 14k
5 votes
3 answers
479 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: $$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\...
CommunityBot
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11 votes
1 answer
1k views
New method to compute square roots [closed]
In 2011 when I was in school I created a formula to calculate square roots... For $x\in\mathbb{R}$ with $x>0$ the following holds: $$\sqrt{x} = \sum_{n=0}^{\infty}\frac{\left(\prod_{k=1}^{n}\left(\...
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2 votes
2 answers
661 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 ...
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0 votes
1 answer
257 views
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 ...
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3 votes
0 answers
361 views
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). $$\int_0^{\infty} \frac{x^{\alpha-1}}{\sqrt{(a+x)^2+z^2}}K(\frac{2\sqrt{...
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3 votes
4 answers
617 views
Some Log integrals related to Gamma value
Two years ago I evaluated some integrals related to $\Gamma(1/4)$. First example: $$(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{...
CommunityBot
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5 votes
3 answers
414 views
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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0 votes
0 answers
124 views
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,-...
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-2 votes
1 answer
211 views
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 $...
1 vote
0 answers
244 views
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
0 answers
104 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 ...
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5 votes
0 answers
502 views
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 &...
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1 vote
0 answers
137 views
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
0 votes
0 answers
185 views
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: $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s) $$ $$\xi(s) = \xi(1 - s)$$ $$\pi^{-s/2}\ \Gamma\left(\...
- 1,203
22 votes
4 answers
10k views
Who invented the gamma function?
Who was the first person who solved the problem of extending the factorial to non-integer arguments? Detlef Gronau writes [1]: "As a matter of fact, it was Daniel Bernoulli who gave in 1729 the ...
- 4,793
6 votes
2 answers
381 views
Inverse Mellin transform of 3 gamma functions product
I want to calculate the inverse Mellin transform of products of 3 gamma functions. $$F\left ( s \right )=\frac{1}{2i\pi}\int \Gamma(s)\Gamma (2s+a)\Gamma( 2s+b)x^{-s}ds$$ Above contour integral has ...
- 1,221
2 votes
2 answers
798 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 $\...
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