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Questions tagged [gamma-function]

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

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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(\...
polygamma's user avatar
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 ...
Joegin 's user avatar
0 votes
1 answer
133 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
0 answers
180 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{...
r-nishi's user avatar
  • 31
3 votes
4 answers
486 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
0 votes
0 answers
48 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
64 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
0 votes
1 answer
158 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 $...
Erik Satie's user avatar
5 votes
3 answers
254 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
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 ...
igorf's user avatar
  • 475
5 votes
0 answers
390 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
  • 51
1 vote
0 answers
50 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 ...
Aschie4589's user avatar
0 votes
0 answers
130 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
  • 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 ...
Pouya's user avatar
  • 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 $\...
velicci's user avatar
  • 29
2 votes
0 answers
98 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
1 vote
1 answer
157 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
275 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
91 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
150 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
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 (...
TPC's user avatar
  • 622
1 vote
1 answer
85 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
577 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
246 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
  • 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 $0<Re(z)<1$;...
user363337's user avatar
1 vote
1 answer
105 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
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, ...
qifeng618's user avatar
  • 706
4 votes
1 answer
404 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
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 ...
searp's user avatar
  • 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 ...
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
317 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
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 ...
Anixx's user avatar
  • 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{...
italiangoat's user avatar
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 $z$ ...
Priyadarshi Paul's user avatar
8 votes
2 answers
905 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
228 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
  • 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 ...
Dianbin Bao's user avatar
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?
user155294's user avatar
-1 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 $t >0$ ...
zeraoulia rafik's user avatar
7 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 ...
Descartes Before the Horse's user avatar
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 ...
James Propp's user avatar
  • 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)}\...
user124297's user avatar
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{...
Jon's user avatar
  • 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, $$\prod_{i=0}^{N}\big(i!\big)=G(N+2)$$ ...
Wiliam's user avatar
  • 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(...
VS.'s user avatar
  • 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\...
VS.'s user avatar
  • 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 ...
William's user avatar
  • 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 ...
William's user avatar
  • 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 $d=3$) ...
leonbloy's user avatar
  • 298