Questions tagged [gamma-function]

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

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Are the twin primes the only positive double zeros of this real function?

Agno's answer was extremely helpful. For $x \in \mathbb{R}, x \ge 1$ define $$ f(x) = \sin\frac{\pi(\Gamma(x)+1)}{\lfloor x \rfloor}$$ By Wilson's theorem the positive integer zeros of $f(x)$ are ...
joro's user avatar
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Do all complex zeros in the strip of $\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$ lie on the critical line?

Numerical evidence suggests that the complex zeros of: $$f(s):=\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$$ all reside on the line $\Re(s)=\frac12$, except for a finite few outside ...
Agno's user avatar
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8 votes
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What is the $q$-analog of $\Gamma(z)\Gamma(1-z)=\frac\pi{\sin(\pi z)}$?

I would expect the $q$-Gamma function to have the property which would be the $q$-analog of the Euler reflection formula from my question title. More concretely: $\Gamma(z)$ has simple poles at ...
მამუკა ჯიბლაძე's user avatar
6 votes
0 answers
212 views

Relation between Gross-Koblitz and Chowla-Selberg formulas

The Chowla-Selberg formula relates the eta function with values of the gamma function at rational numbers. The eta function appears, at least in the proofs I have seen, related to values of $L$-...
efs's user avatar
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5 votes
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414 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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645 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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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
5 votes
0 answers
466 views

integrating with respect to parameters in beta function

I would like to evaluate an integral: $$\int_t^1\frac{1}{B(1+s\phi,1+\phi(1-s))}p^{s\phi}(1-p)^{\phi(1-s)}ds,$$ where $B(a,b)$ is a beta function and $p\in(0,1)$ and $\phi>0$ are some parameters. ...
Joanna F's user avatar
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how understand periodicity in a combination of power, gamma and zeta functions?

Riemann's functional equation may be written: $$ \frac{\zeta(s)}{\zeta(1-s)} = 2^s \pi^{s-1} \sin(\frac{\pi s}2) \Gamma(1-s) \tag{1} $$ and so by symmetry: $$ \frac{\zeta(1-s)}{\zeta(s)} = 2^{1-s} \pi^...
David Holden's user avatar
5 votes
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400 views

A relation between the Gamma function and the Mobius function?

It is well known how altering the integral for the Gamma function: $$\displaystyle \Gamma(s) = \int_0^\infty t^{s-1} e^{-t}\,dt$$ through substituting $t=nx$, $$\displaystyle \Gamma(s)\frac{1}{n^s} ...
Agno's user avatar
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5 votes
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The Riemann Zeta Function summing over the Gamma Function

Has anyone studied a function of the form $$\eta(s) = \sum_{n=1}^{\infty} \frac{1}{\Gamma(n)^{s}} = \sum_{n=0}^{\infty}\frac{1}{k!^s}$$ This series is appearing in my research on the volumetric ...
Samuel Reid's user avatar
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4 votes
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183 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
3 votes
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251 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
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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
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216 views

Gosper's Beta function identities

According to the last paragraph in Mathworld's Beta function article http://mathworld.wolfram.com/BetaFunction.html, Gosper found some multiplication formulas for the Beta function, but it does not ...
efs's user avatar
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3 votes
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Reflection Formulas for the $\Gamma$ Function

We have $$\begin{align} &\Gamma\Big(1~+~x\Big)~\cdot~\Gamma\Big(1-x\Big)~=~\frac{\pi x}{\sin\pi x} \\\\ &\Gamma\Big(\tfrac12+x\Big)~\cdot~\Gamma\Big(\tfrac12-x\Big)~=~\frac\pi{\cos\pi x} \\\...
Lucian's user avatar
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Generalization of Trigonometric, Hyperbolic, $\Gamma$ and $\zeta$ Functions

It is becoming increasingly clear that the expression $~\displaystyle\prod_{n\in\mathbf M}\left[1-\frac{z^s}{(n-a)^s}\right],~$ with $~|\mathbf M|=\aleph_0,~$ $a\not\in\mathbf M,~$ and $~\Re(s)>1,~$...
Lucian's user avatar
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3 votes
0 answers
565 views

Sum of binomial coefficients weighted by a lower incomplete regularized gamma function

The following summation turned up in the course of my research: $$S_n=\sum_{k=0}^n {n \choose k}\lambda^k P(k,t)$$ where $P(k,t)=\frac{1}{\Gamma(k)}\int_{0}^t e^{-x}x^{k-1}dx$ is the lower ...
Bullmoose's user avatar
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Derivative of the regularized upper incomplete gamma function

I have a question about the derivative of the regularized upper incomplete gamma function. Considering the gamma function and the incomplete gamma function \begin{eqnarray} \Gamma(x)&=&\int_0^\...
ppyang's user avatar
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264 views

Alternating sums of powers of the lngamma ($\small f_p(x) = \sum \log(k!)^p x^k $ at $\small x=-1$)

I'm still fiddling with this recent question and come to a detail, whether I can find closed forms for the sums of the $\small lngamma() $ function. Precisely $$ \small f_p(x) = \sum_{k=0}^{\infty} \...
Gottfried Helms's user avatar
3 votes
0 answers
170 views

Iterated Kumaraswamy distributions

The Kumaraswamy distribution has cdf $F(x;a,b) = 1-(1-x^a)^b$. Does anyone know any formulas or properties relating to iterations of this on itself, meaning $$ F_i(x;a,b) = 1-(1-F_{i-1}^a)^b$$ If ...
OctaviaQ's user avatar
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2 votes
0 answers
117 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
2 votes
1 answer
327 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
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
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2 votes
0 answers
143 views

Monotone coupling between "two-sided Gumbel" distributions

I am interested in finding a monotone coupling between two random variables. Let $\alpha_1>\alpha_2$, $b<a$. Define the following two (non-normalized) densities on the whole real line: \begin{...
Leonid Petrov's user avatar
2 votes
0 answers
219 views

Generalized hypergeometric function at $z=1$

I wonder if there is a closed formula for the following generalized hypergeometric function at $z=1$: $${}_4F_3\left(a,a,a,a;a+1,a+1,2a;1\right)$$ Specifically, I would like to have a formula in ...
D M's user avatar
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0 answers
745 views

About the Stirling's approximation of the incomplete gamma function $\left|\gamma\left(a+ib,z\right)\right|$

Let $a+ib$ be a complex number. It is well-known that $$\left|\Gamma\left(a+ib\right)\right|\sim\sqrt{2\pi}e^{-\pi\left|b\right|/2}\left|b\right|^{a-1/2}$$ for any fixed $a$ and $\left|b\right|\...
User's user avatar
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2 votes
0 answers
102 views

Gamma function in terms of a linear function

I have noticed that the plot of the function $[\Gamma(x+1)]^{1/x}$ for $x > 0$ looks like a straight line at all scales. This implies that $\Gamma(x+1) \approx ((1-e^{-\gamma})x+e^{-\gamma})^x$ for ...
Ivan Matychyn's user avatar
2 votes
0 answers
201 views

Are all zeros of $\dfrac{\Gamma'}{\Gamma^2}(s) \pm \dfrac{\Gamma'}{\Gamma^2}(1-s)$ either real or on the line with $\Re(s)=\frac12$?

In this post it was proven that, except for a finite few, all zeros of $\Gamma(s) \pm \Gamma(1-s)$ are either real or reside on the line $\Re(s)=\frac12$. I have been searching for similar reflexive $...
Agno's user avatar
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2 votes
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161 views

Are all complex zeros of $\Psi^{(s)}(1) \pm \Psi^{(1-s)}(1)$ on the critical line?

The balanced polygamma function $\Psi^{(s)}(x)$ for $x=1$ can be expressed as: $$\Psi^{(s)}(1)=\dfrac{\big(\Psi(-s)+\gamma\big)\,\zeta(s+1)+\zeta'(s+1)}{\Gamma(-s)}$$ Note $\Psi(s)$ is the digamma ...
Agno's user avatar
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2 votes
0 answers
357 views

Recurrence formula for digamma function with rational number

It is well known that $\psi\left( x+N\right) =\psi\left( x\right) +\sum_{k=0}^{N-1}\frac{1}{x+k}$. Is there a recurrence formula for $\psi\left( x+\frac{p}{q}\right) $ where $\frac{p}{q}$ is ...
MotiLevy's user avatar
2 votes
0 answers
293 views

Coutour Integral of Gamma Functions

How do I solve the Integral $$ \frac{1}{2\pi j} \oint \frac{b^{ - s} \Gamma[2 + i - s] \Gamma[s] \Gamma[-1 - i + s]}{ (2 + i - s) \Gamma[3 + i - s]} \:\mathrm{d}s$$ This integral is an inverse ...
Remy's user avatar
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2 votes
0 answers
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Can any antidifference (indefinite sum) of a function be expressed in elementary functions and generalized polygamma function if its integral can be expressed in elementary functions?

If the integral or multiplicative integral of a function can be expressed with elementary functions, does it mean its indefinite sum (antidifference) or indefinite product respectively can be ...
Anixx's user avatar
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1 vote
0 answers
26 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
1 vote
0 answers
114 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 ...
Andrew's user avatar
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1 vote
0 answers
144 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 vote
0 answers
67 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
1 vote
1 answer
289 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
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1 vote
0 answers
108 views

The complex roots of $\left(\Gamma(\kappa s)+2^{1-s} \pi^{-s}\cos\left(\frac{\pi s}{2} \right)\Gamma(s)\,\Gamma(\kappa(1-s))\right)$

Question: With $\kappa \in \mathbb{R}, \kappa \ge 1$, the function: $$\Upsilon(s,\kappa) = \Gamma(\kappa s)\zeta(s)+\Gamma\left(\kappa(1-s)\right) \zeta(1-s)$$ could be rewritten as: $$\Upsilon(s,\...
Rudolph's user avatar
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1 vote
0 answers
72 views

The integral of $\Gamma\left(\zeta\right) \, W_{-\zeta,\mu}(z) $

Someone has a reference that addresses an integral of the followns type $$I_{a,b,x} = \int_{0}^{+\infty} \zeta^{-a} \, \Gamma\left(\zeta+b\right) \, W_{-\zeta-b,\tfrac{-1}{2}}(x) \, d\zeta$$ where ...
Z. Alfata's user avatar
  • 640
1 vote
0 answers
1k views

Proving injectivity of a multivariable function

Let $I$ denote the interval $(0,\infty)$, we define the function $f:I^2\to I^2$ by, $$f(x,y)=\left({\Gamma(4x+y)\Gamma(y)\over {\Gamma(2x+y)}^2},{\Gamma(4x+y)\Gamma(2x+y)\over {\Gamma(3x+y)}^2}\right)$...
LayZ's user avatar
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1 vote
0 answers
260 views

Convergence of $\sum_{n=1}^\infty \frac{\psi^{(1)}(1-n/\pi)}{n^3}$

Related to an open problem about another series. Set $$A= \sum_{n=1}^\infty \frac{\psi^{(1)}(1-n/\pi)}{n^3}$$ where $\psi^{(n)}(k)$ is the polygamma function. Does $A$ converge? The related ...
joro's user avatar
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1 vote
0 answers
522 views

Poles of products of Gamma functions

I want to know if there can be a general statement about the poles (Laurent expansion) of such products of Gamma functions as a function of $p \in \mathbb{R}$ in the limit $\epsilon \rightarrow 0$, $\...
Anirbit's user avatar
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1 vote
0 answers
332 views

Does $e^{az}/\Gamma(z)dz$ have a nice indefinite integral? Definite integral?

Now obviously we can just expand $\frac{1}{\Gamma(z)}$ into a power series and then integrate with $e^{az}$. But the coefficients of $\frac{1}{\Gamma(z)}$ are way too ugly. We can represent $\frac{1}...
SorcererofDM's user avatar
1 vote
0 answers
299 views

Integrating gamma products and quotients over a vertical line

The Wolfram functions collection contains a small number of integrals of products and quotients of terms $\Gamma(a_i\pm t)$ over a vertical line, all of which can be evaluated in terms of only gamma ...
Wolfgang's user avatar
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0 votes
0 answers
69 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
0 votes
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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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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0 votes
0 answers
121 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
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