Questions tagged [gamma-function]
used only for functions based on gamma, not functions with some obscure relation to gamma
182 questions
1 vote
0 answers
145 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 ...
- 11
0 votes
0 answers
61 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: $$ B(x, y) \sim \sqrt{2\pi} \cdot \frac{x^{x - \frac{1}{2}...
- 687
0 votes
0 answers
129 views
Asymptotic behaviour of the analytic integral
Given $m \ge 0$ and $n$ odd, $ s >0 $ is there a way, for computing asymptotic expansion of following integral: $$ \int_{\mathcal{C}} \int_{\mathcal{C}} \frac{ \Gamma(m + s_1) \Gamma(m + s_2) \...
- 1
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
160 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
9 votes
1 answer
526 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 ...
- 13.9k
2 votes
1 answer
238 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 ...
- 1,564
2 votes
0 answers
123 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 $n^\text{th}$ ...
- 21
0 votes
0 answers
214 views
How to prove the convergence of the following series involving Gamma function?
Consider the following result（$d$ denotes the dimensions and $0<t<T$） $$c\left(\sum_{j=0}^\infty\frac{\Gamma^j(1-\kappa)}{\Gamma((j+1)(1-\kappa))}t^{j(1-\kappa)-\kappa}\right)^{\frac{1}{2}}\leq ...
- 57
1 vote
3 answers
298 views
The approximated function of $\mathbb{E}\left\{ \ln\left(1+X\right)\right\}$, where $X\sim\operatorname{Gamma}\left(\kappa\ge1,\theta>0\right)$
Given $X \sim \operatorname{Gamma}(\kappa, \theta)$ with CDF $F_X(\kappa, \theta)$, where $\kappa \geq 1$ and $\theta > 0$, the expected value of $\mathbb{E} \left\{ \ln(1+X) \right\}$ is ...
- 61
0 votes
0 answers
160 views
how to use Gauss Multiplication Formula for Gamma function?
I asked this question in MSE I studied Gauss Multiplication Formula which known for $n\in Z^+ \wedge nx\notin Z^-\cup\{0\}$ $$\Gamma(nz)=(2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\Gamma\left(z+\...
- 711
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
155 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^{...
- 271
1 vote
1 answer
280 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 ...
- 21
7 votes
1 answer
942 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 ...
- 697
4 votes
2 answers
293 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 $$\eta (\tau)=e^{\pi i\tau/12}\prod_{n=1}^\infty (1-e^{2\pi i n\tau}),\quad \Im\tau\...
- 317
3 votes
1 answer
713 views
Separating Gamma in two independent functions
I've encountered a problem in my PhD. I would greatly appreciate any suggestions, tips, or comments you might have. The problem is Let $\Gamma(s,x)$ be the incomplete gamma function. Given integers $n ...
- 41
1 vote
0 answers
60 views
Exponential-like function equivalent for the Dixonian Elliptics
Is there some exponential-like function that acts as partner for the intricate Dixonian Elliptics, in a similar way that the Exponential function acts as a partner for the trigonometric functions ?
5 votes
1 answer
800 views
How to evaluate inverse Laplace transform of $e^{- \sqrt{s}} $ using series?
I tried to find an inverse Laplace transform by series as follows $$ 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)$$ and by ...
- 711
1 vote
1 answer
233 views
T functions arising from derivatives of incomplete Gamma function
Here the derivatives of the incomplete gamma functions are described via: $$ 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|...
- 13
0 votes
1 answer
177 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 ...
- 402
1 vote
1 answer
130 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 ...
- 583
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 ...
- 1,099
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....
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: $$\sqrt{x} = \sum_{n=0}^{\infty}\frac{\left(\prod_{k=1}^{n}\left(\...
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2 votes
2 answers
659 views
Integral calculus with Gamma function [closed]
I have to prove that for $0<\alpha<1$ and $\beta>0$, \begin{equation} \int_{0}^{\infty} x^{-\alpha-1}\left(e^{-\beta x}-1\right)dx=\beta^\alpha\Gamma(-\alpha), \end{equation} and I have ...
- 95
0 votes
1 answer
257 views
Infinite limit of sums of gamma functions is constant?
The following expression arises in the study of hierarchical models. I suspect that the sum of the underlined $4$ terms become constant as $\alpha\rightarrow \infty$. Mathematica agrees when prompted ...
- 111
3 votes
0 answers
359 views
Derivation of an integral containing the complete elliptic integral of the first kind
I found the following formula in "INTEGRALS AND SERIES, vol.3" by Prudnikov, Brychkov and Marichev (page 188, eq.5). $$\int_0^{\infty} \frac{x^{\alpha-1}}{\sqrt{(a+x)^2+z^2}}K(\frac{2\sqrt{...
- 31
3 votes
4 answers
613 views
Some Log integrals related to Gamma value
Two years ago I evaluated some integrals related to $\Gamma(1/4)$. First example: $$(1)\hspace{.2cm}\int_{0}^{1}\frac{\sqrt{x}\log{(1+\sqrt{1+x})}}{\sqrt{1-x^2}} dx=\pi-\frac{\sqrt {2}\pi^{5/2}+4\sqrt{...
user497425
0 votes
0 answers
114 views
Incomplete Gamma function $\Gamma(0,x)$ and $\Gamma(0,-x)$
I want to find the value of this \begin{align} y=\Gamma(0,x)-\Gamma(0,-x) \end{align} where $\Gamma$ is the upper incomplete Gamma function, $x>0$ is real. I can't find the definition of $\Gamma(0,-...
- 11
1 vote
0 answers
244 views
polynomial approximation of hypergeometric function 2F1
I have the following function $T(k_1,k_2)$ resulting from multiphoton transition matrix elements calculations: $T(k_1,k_2)=\gamma^{-k_2}\sum_{j=0}^{k_1}(j+2)_{l+1}\binom{k_1}{j}(k_1+1)_3(\gamma-1)^{j}{...
- 11
-2 votes
1 answer
211 views
Conjecture about the equality : $f(y)=y\ln(y)+\sum_{n=2}^{\infty}\pm\frac{\left(y\ln y\right)^n}{2^{a_n}}$
I try here because I expect I cannot have any answer on MSE : Problem : Let : $$f\left(x\right)=\frac{\left(1+\ln27\right)x!\ln x!}{x+1},g\left(x\right)=x\ln x$$ Then it seems $\exists y\in(0,1)$ and $...
5 votes
3 answers
410 views
Evaluating the series $\sum_{n=0}^{\infty} n! x^n$ and inverse variable-fractional-derivatives
So I was interested in formally assigning values to the completely divergent series $G(x) = \sum_{n=0}^{\infty} n!x^n $. I guess the question COULD end here if you already have an idea of how to ...
- 3,139
0 votes
0 answers
104 views
An interpolation of $n!$ such that its derivatives have few zeros
The $\Gamma$-function restricted to $(0,+\infty)$ has the following properties: $\Gamma(n)=(n-1)!$ for $n=1,2,3,...$. The $k$'th derivative $\Gamma^{(k)}$ has no zeros on $(0,+\infty)$ when $k$ is ...
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5 votes
0 answers
500 views
Determinant of Hankel matrix with $a_n=(n!)^2$
Consider a Hankel matrix of the form $H_n(a_0(n))=\begin{pmatrix} a_0(n) & (1!)^2 & (2!)^2 & \cdots & (n!)^2\\ (1!)^2 & (2!)^2 & (3!)^2& \cdots & ((n+1)!)^2\\ (2!)^2 &...
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1 vote
0 answers
137 views
Uniform bound on product of Gamma functions in an article by Jerison and Kenig
I am new here. I'm reposting a question I originally posted here on Math Stack Exchange. I realized that maybe this is more appropriate place to ask such a question... I have been trying to read ...
- 11
0 votes
0 answers
185 views
Does it make sense to express upper bounds on arithmetic sequences with Dirichlet generating functions?
In order to see what happens when taking the functional equation in this form: $$\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
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 ...
- 69
2 votes
2 answers
793 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 $\...
- 39
2 votes
0 answers
178 views
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
2 votes
1 answer
414 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: $$ \frac{\Gamma(s/2)^2}{2^...
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5 votes
2 answers
501 views
Extended binomial coefficients and the gamma function
For which $(a,b,n) \in \mathbb{Z}^3$ satisfying $a+b=n$ does $\frac{\Gamma(z+1)}{\Gamma(x+1)\Gamma(y+1)}$ approach a limit as $(x,y,z) \rightarrow (a,b,n)$ in $\mathbb{C}^3$, and what is that limit? (...
- 20.1k
0 votes
0 answers
190 views
Addition formulas for q-analogs of trigonometric functions
Sine and Cosine functions possess notable formulas for addition of angles $$ \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). $$ One can ...
- 158
2 votes
1 answer
258 views
Gamma function and the somewhat extended version of Bohr-Mollerup theorem
The Gamma function $\Gamma$ is defined by \begin{equation*} \Gamma(x)=\int_{0}^\infty t^{x-1}e^{-t} \,\mathrm{d}t, \end{equation*} for $x>0$. It satisfies the well-known functional equation $$\...
- 143
5 votes
0 answers
686 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
142 views
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 $$\sum_{m=0}^\infty \frac{\text B_z(m+a,b-m)x^m}{m!}=\sum_{m=0}^\infty z^{m+a}\sum_{n=0}^\infty\frac{(1-(b-m))...
- 240
5 votes
0 answers
790 views
The Basel problem revisited?
In the Basel problem, the $sinc$ function is considered at the Wikipedia page. Let me try to make an alternative function definition: $$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \...
- 3,347
4 votes
2 answers
417 views
$\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} z^{2k}$ 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{...
- 322
2 votes
1 answer
501 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)}...
- 29
1 vote
1 answer
176 views
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 ...
