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3
votes
1answer
214 views

Do the roots of this equation involving two Euler products all reside on the critical line?

This question loosely builds on the second part of this one. Take the Riemann $\xi$-function: $\xi(s) =\frac12 s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s)$. Numerical ...
3
votes
1answer
114 views

Xi Function on Critical Strip - Mellin Transform

Story I'm trying to prove following identity $$\int_0^\infty \frac{\Xi(t)}{t^2 + \frac{1}{4}} \cos(xt) dt = \frac{1}{2} \pi (e^{\frac{1}{2}x} - 2e^{-\frac{1}{2}x} \psi(e^{-2x}))$$ where $$\psi(x)=\...
3
votes
2answers
352 views

An integral identity evaluating the gamma function

While reading a number theory paper I encountered the identity $$ \int_{- \infty}^{\infty} (1 + x^2)^{ - \frac{z}{2} - 1} dx = \sqrt{\pi} \frac{ \Gamma(\frac{z + 1}{2}) }{\Gamma(\frac{z}{2} + 1)},$$ ...
1
vote
0answers
43 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 ...
1
vote
0answers
65 views

For which primes $p$ is the field $\mathbb{Q}(\Gamma(1/p^{j}))$ a strict subfield of $\mathbb{Q}(\Gamma(1/p^{i}))$ whenever $0<i<j$?

I already asked this question on a French math forum but eventually came to think that as silly it may turn out to be, perhaps something interesting could finally emerge from it, so I decided to take ...
2
votes
2answers
76 views

Solution of $\left(\Gamma(x+c)/ \Gamma(x+d)\right)y^d/y^c = {\rm const}$

When I try to solve $F(x,y)= \Gamma(x+c)/\Gamma(x+d) y^d/y^c = {\rm const}$, I find that $y = p x + q$ satisfies the above equation, whith specific $p$ and $q$ constants for the given constants $c$ ...
2
votes
1answer
142 views

Proof of Euler's reflection formula via rapidly decreasing Fourier series

Story I want to prove Euler's reflection formula by showing that \begin{equation*} f(s) = \sin(\pi s) \Gamma(s) \Gamma(1 - s) \end{equation*} is constant, where $s = \sigma + it$. It's easy to see ...
2
votes
1answer
86 views

Estimating $\left(\Gamma\left(\frac{\alpha}{\sqrt n}\right)\right)^n$ for fixed $\alpha >0$ as a function of (large) $n$ [closed]

I am interested in the growth of $$f_\alpha(n)=\left(\Gamma\left(1+\frac{\alpha}{\sqrt{n}}\right)\right)^n,$$ for positive $\alpha$ and large $n$. (Preferably something more precise than $1! \leq f_\...
84
votes
10answers
11k 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 ...
9
votes
3answers
435 views

Complex proof of $B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$

It is a question in spirit of this one. Is there a way to prove Euler's formula $$ \int_0^1 x^{a-1}(1-x)^{b-1}dx=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} $$ using contour integration (and maybe ...
3
votes
1answer
242 views

Integral involving the gamma function

If we define $$f(x) = 1 + \frac{\cos\big(\pi\frac{\Gamma(x) + 1}{x}\big)}{2 - \cos(2\pi{x})}$$ how would one go about evaluating $$ \int_1^R \frac{1}{x} \log{f(xe^{i\alpha})} dx$$ for some parameter ...
10
votes
1answer
170 views

Probability distribution derived from gamma function - does it have a name?

Consider the complex gamma function, denoted by $\Gamma(\sigma+it)$. Now, let's fix $\sigma$ and let t vary. Then consider the following expression: $$|\Gamma(\sigma+it)|^2$$ For any choice of $\...
7
votes
2answers
321 views

Calculation of integral using Gamma function when the imaginary part is zero

Consider the following expression of Gamma function $$\frac{\Gamma(z)}{p^z}=\int_{0}^{\infty}e^{-pt}t^{z-1}dt \ \ \ \ \ \ \ \ \ (1)$$ where $Re(z)>0$ and $Re(p)>0$. In Lebedevs book "special ...
14
votes
2answers
567 views

A conjecture about $\lfloor n!\cdot q/e\rfloor-\,!n\cdot q$

I was thinking about this question asked at Math.SE, when I came up with the following conjecture. For every $q\in\mathbb Q$ consider a sequence $s_n^{(q)}$ (terms within the sequence are indexed by $...
5
votes
1answer
145 views

On a Sum of Gamma Functions

I am working on a problem where the following sum appears: $$F(s, t)=\frac{1}{\Gamma(1+2\alpha)}\sum_{n=0}^{\infty}{\frac{s^{n} t^{n}}{\left[(s+1)(t+1)\right]^{n+1+\alpha}}\frac{\Gamma(n+1+2\alpha)}{\...
2
votes
0answers
140 views

Continued Fraction: Please prove $\frac{1}{e \gamma (x+1,1)}=x+\frac{1}{x+1+\frac{2}{x+2+\frac{3}{x+3+\frac{4}{\dots}}}}$ [closed]

I have been playing around with Mathematica and continued fractions and I noticed something. ContinuedFractionK[n, n + x, {n, 1, Infinity}] ==-x + 1/(E Gamma[1 + x] - E Gamma[1 + x, 1])==-x + 1/(E ...
8
votes
2answers
584 views

Inequality for a gamma function

Let $s=\sigma+it$ and $\Gamma(s)$ be the Euler gamma function. Does the inequality hold? $$ \left|\frac{\Gamma(s)}{\Gamma(2-s)}\right|\leq |s|^{2(\sigma-1)},\, 1<\sigma<2,\, t\in \mathbb{R}. $$...
1
vote
0answers
143 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)$...
3
votes
2answers
364 views

Simplifying the expression involving instances of Gamma function

Is it possible to simplify the following expression involving instances of Gamma function: $$E(p)=\frac{\frac{\Gamma(\frac{p+1}{2})}{\Gamma(\frac{p+2}{2})}} {\left(\frac{\Gamma(\frac{p+1}{p})^2}{\...
6
votes
0answers
210 views

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 ...
6
votes
1answer
139 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) \...
2
votes
0answers
96 views

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

veThe 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 ...
3
votes
0answers
65 views

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^...
9
votes
0answers
324 views

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 ...
4
votes
0answers
247 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} ...
19
votes
3answers
695 views

Is this combination of generalized polygamma and dilogarithm actually zero? $\Im\;\psi^{(-2)}(1+i)+\frac1{4\pi}\text{Li}_2(e^{-2\pi})-\log\sqrt{2\pi}+\frac{5\pi}{24}+\frac12$

I encountered this quantity in my calculations and tried to simplify it. Approximate numeric calculations suggested it could be zero (more precisely, it is certainly less than $10^{-4\times10^3}$ in ...
1
vote
0answers
101 views

How naturally can functions defined by parametric integrals be interpolated from $\mathbb N$ to $\mathbb R^+$?

It is well known that several definite integrals $I_n$ containing a parameter $n\in\mathbb N$ can be expressed recursively (e.g. doing integration by parts) in terms of $I_{n-1} $ or $I_{n-2} $, and ...
8
votes
1answer
879 views

Algorithm to produce random number with a gamma distribution

I'd like to produce pseudo-random numbers with different distributions for a Monte Carlo simulation. I've got the poisson distribution working nicely with an algorithm from Knuth. I'm having trouble ...
11
votes
0answers
701 views

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 ...
6
votes
2answers
1k views

Multiplicative integral of $\Gamma(x)$

A recent question on the notion and notation of multiplicative integrals ( What is the standard notation for a multiplicative integral? ) induced me to play with the Riemann products of the Gamma ...
6
votes
1answer
274 views

Conjectured bound on Kummer's function (confluent hypergeometric function)

I believe the following bound to be correct: ${}_1F_1(-a;1+a;-z) a z^{-a} \gamma(a,z) \leq 1$ for real-valued a > 0 and z $\geq 0$. $\gamma(a,z)$ is the lower incomplete gamma function. Apart from ...
9
votes
5answers
843 views

$\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$

I'm trying to solve the integral $\int^{\infty}_{0}x^{r +s- 1}(1 + x)^{-s}(1 + x^2)^{-\frac{rm}{2}}dx$, where $s$, $r$ and $m$>1 are positive integers. My question is whether a closed form ...
1
vote
0answers
137 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 ...
2
votes
0answers
290 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 ...
1
vote
0answers
228 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 ...
0
votes
3answers
276 views

Inequality for the modulus of Riemann zeta on horizontal lines and alleged partial result of Maple

According to a conjecture p.4 $|\zeta(\frac12 -\Delta + it))| > |\zeta(\frac12 + \Delta + i t|$ for $0 < \Delta < \frac12$ and $|t| > 2 \pi +1$. Since $\zeta(\overline{s}) = \overline{\...
3
votes
0answers
238 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} \...
45
votes
6answers
3k views

Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real part = $\frac12$ ?

The function $\Gamma(s)$ does not have zeros, but $\Gamma(s)\pm \Gamma(1-s)$ does. Ignoring the real solutions for now and assuming $s \in \mathbb{C}$ then: $\Gamma(s)-\Gamma(1-s)$ yields zeros at: ...
5
votes
0answers
337 views

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 ...
11
votes
1answer
324 views

Can a harmonic number be a rational number for non-integer rational argument?

Define harmonic numbers for a complex argument $z$ as $H_z=\frac{\Gamma'(z+1)}{\Gamma(z+1)}-\Gamma'(1)$. For $n\in\mathbb{N}$, $H_n$ are usual harmonic numbers $\sum^n_{k=1} k^{-1}$ . They are ...
2
votes
1answer
188 views

Growth of the reciprocal gamma function in the critical strip

I was wondering if there are any results that studied the growth of $\left|\frac{1}{\Gamma(s)}\right|$ where $0 < \Re(s) < 1$ and as $\Im(s) \to \infty$? Any pointers to any results, papers, ...
1
vote
0answers
306 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 $\epsilon \rightarrow 0$, $\...
2
votes
0answers
233 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 ...
4
votes
1answer
547 views

Is there a closed form expression/series expansion for $\int_{\epsilon-i\infty}^{\epsilon+i\infty} e^{az+b^2z^2}\Gamma(z)\Gamma(1-z)dz$ ?

I've been trying to find a closed form expression/series expansion for the following integral without success: $$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} e^{az+b^2z^2}\Gamma(z)\Gamma(1-z)dz$$...
2
votes
0answers
1k views

Derivative of the regularized upper incomplete gamma function

Hello everyone! 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)&...
5
votes
4answers
521 views

Integral transform and $\frac{1}{n!}$.

Probably this is a trivial question, but I am unable to find an answer: is there a function $v(x)$ such that $$ \int_{0}^\infty x^n e^{v(x)} dx =\frac{1}{n!} $$ for all positiv integer n?
1
vote
0answers
264 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}...
3
votes
1answer
620 views

Connection between Bernoulli polynomials and polygamma function

There is an intricate connection between Hurwitz Zeta and the (traditional) polygamma function: $$\psi_n(z)=(-1)^{n+1}n!\zeta(n+1,z)$$ If to use a generalization for Bernoulli numbers, this can be ...
4
votes
1answer
841 views

q-Pochhammer Symbol Identity

Is this identity or an equivalent one already referenced in the litterature? In particular, is it even true? ${\frac{\left ( -1 ; e^{-4\pi} \right) ^2_{\infty}}{\left ( e^{-2\pi} ; e^{-2\pi} \right) ^...
5
votes
1answer
313 views

On the multidimensional generalisation of Gamma function

Gamma function is defined as $$ \Gamma(z) = \int\limits_{0}^{+\infty} x^{z-1} e^{-x} \; dx $$ I'm looking for multidimensional generalisation of this definition. I consider the class $Q$ of positive,...