Questions tagged [gamma-function]

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

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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\...
Nomas2's user avatar
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1 answer
590 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 ...
curiosity96's user avatar
-2 votes
1 answer
189 views

An equality between $\pi$ and $\Gamma$ function [closed]

Consider the following equality: $$\sum_{n=1}^{+\infty} (-1)^{n+1} \frac{(\frac{(2n-3)!!}{(2n-2)!!})^2*\frac{\pi}{2}}{n}=\frac{\Gamma(\frac{1}{4})^2}{2\sqrt{2\pi}}-\frac{2\sqrt{2}*\pi^{\frac{3}{2}}}{\...
Craw Craw's user avatar
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29 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 ?
Jaime Yerbabuena's user avatar
5 votes
1 answer
272 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 ...
Faoler's user avatar
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1 answer
176 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|...
user18722294's user avatar
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1 answer
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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 ...
Bobby Ocean's user avatar
1 vote
1 answer
80 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 ...
Hvjurthuk's user avatar
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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 ...
Andrew's user avatar
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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....
Felipe Augusto de Figueiredo's user avatar
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(\...
polygamma's user avatar
2 votes
2 answers
579 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 ...
Joegin 's user avatar
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1 answer
183 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 ...
cataclysmic's user avatar
3 votes
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256 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
527 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{...
user avatar
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0 answers
70 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,-...
Charlie Nie's user avatar
1 vote
0 answers
149 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}{...
Omer Amit's user avatar
-1 votes
1 answer
178 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 $...
DesmosTutu's user avatar
5 votes
3 answers
321 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 ...
Sidharth Ghoshal's user avatar
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0 answers
78 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 ...
igorf's user avatar
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5 votes
0 answers
420 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 &...
fs98's user avatar
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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 ...
Aschie4589's user avatar
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0 answers
140 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(\...
Mats Granvik's user avatar
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5 votes
2 answers
246 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 ...
Pouya's user avatar
  • 59
2 votes
2 answers
578 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 $\...
velicci's user avatar
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2 votes
0 answers
118 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, ...
Mr.MathDoctor's user avatar
2 votes
1 answer
276 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^...
User1326's user avatar
5 votes
2 answers
351 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? (...
James Propp's user avatar
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0 votes
0 answers
123 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 ...
Matteo's user avatar
  • 106
2 votes
1 answer
176 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 $$\...
Mr.MathDoctor's user avatar
5 votes
0 answers
646 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 (...
TPC's user avatar
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1 vote
1 answer
102 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))...
Tyma Gaidash's user avatar
5 votes
0 answers
650 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 ) = \...
mathoverflowUser's user avatar
3 votes
2 answers
275 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{...
Z. Alfata's user avatar
  • 588
2 votes
1 answer
347 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 $0<Re(z)<1$;...
user363337's user avatar
1 vote
1 answer
126 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 ...
George McGonagall's user avatar
1 vote
1 answer
296 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, ...
qifeng618's user avatar
  • 838
4 votes
1 answer
449 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 - ...
AndreaPaco's user avatar
4 votes
2 answers
199 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 ...
searp's user avatar
  • 41
3 votes
0 answers
235 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 ...
Igor's user avatar
  • 31
7 votes
3 answers
1k 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}\...
Carlo Beenakker's user avatar
5 votes
1 answer
387 views

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 ...
matt stokes's user avatar
2 votes
0 answers
56 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 ...
Anixx's user avatar
  • 9,312
2 votes
3 answers
467 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{...
italiangoat's user avatar
2 votes
1 answer
274 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 $z$ ...
Priyadarshi Paul's user avatar
8 votes
2 answers
1k 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. ...
user avatar
5 votes
2 answers
237 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\...
Max M's user avatar
  • 804
4 votes
0 answers
185 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 ...
Dianbin Bao's user avatar
0 votes
0 answers
113 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?
user155294's user avatar
-1 votes
1 answer
175 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 $t >0$ ...
zeraoulia rafik's user avatar