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83
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 ...
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: ...
19
votes
3answers
688 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 ...
14
votes
2answers
2k views

Why does the Gamma function satisfy a functional equation?

In question #7656, Peter Arndt asked why the Gamma function completes the Riemann zeta function in the sense that it makes the functional equation easy to write down. Several of the answers were from ...
14
votes
2answers
2k views

Importance of Log Convexity of the Gamma Function

The Bohr-Mollerup theorem states that the Gamma function is the unique function that satisfies: 1) f(x+1) = x*f(x) 2) f(1) = 1 3) ln(f(x)) is convex The Gamma function is meant to interpolate the ...
14
votes
2answers
565 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 ...
11
votes
1answer
317 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 ...
11
votes
0answers
697 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 ...
10
votes
1answer
164 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 ...
9
votes
5answers
832 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 ...
9
votes
3answers
425 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 ...
9
votes
0answers
311 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 ...
8
votes
3answers
868 views

No simple duplication formula for factorials?

Many special functions including the gamma function have a duplication formula of some sorts. In the case of the gamma function it reads: Gamma(2z) = Gamma(z) Gamma(z+1/2) 22z-1/Gamma(1/2) On ...
8
votes
1answer
779 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 ...
8
votes
1answer
1k views

Errata for Emil Artin's 'The Gamma Function'?

In the English translation of The Gamma Function by Emil Artin (1964 - Holt, Rinehart and Winston) there appears to be a mistake in the formula given for the gamma function on page 24: $$\Gamma(x) = ...
8
votes
2answers
572 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}. ...
7
votes
2answers
280 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 ...
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
132 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) ...
6
votes
1answer
270 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 ...
6
votes
0answers
194 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 ...
5
votes
4answers
520 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?
5
votes
5answers
735 views

Summation of an expression

Hi, Does anyone have an idea about an exact or approximate formulae for the following summation? $$ \sum_{j=1}^n \frac{j^k}{(j-1)!} $$ where k is a positive integer (the denominator of the j^th term ...
5
votes
1answer
134 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} ...
5
votes
1answer
722 views

Gamma function versions of combinatorial identites?

We can extend the binomial coefficient $\binom{n}{k}$ to $\mathbb{R}$ or $\mathbb{C}$ by defining $\binom{x}{y}=\frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma(x-y+1)}$. Do any the standard binomial coefficient ...
5
votes
1answer
304 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 ...
5
votes
0answers
335 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 ...
4
votes
1answer
814 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) ...
4
votes
2answers
544 views

Generalized binomial coefficients and Gaussian density

I ran into an expression calculating the expected value of $\exp(i t \sigma)$ where $\sigma$ is the total number of cycles in a uniformly chosen $S_n$ element. The expression is $$E_n (\exp(i t ...
4
votes
1answer
537 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} ...
4
votes
0answers
244 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} ...
3
votes
3answers
493 views

Reference request for a “well-known identity” in a paper of Shepp and Lloyd

I ran into a "well-known identity" on page 345 of Shepp and Lloyd's On ordered cycle lengths in a random permutation: $$\int_x^{\infty} \frac{\exp(-y)}y dy = \int_0^x \frac{1-\exp(-y)}y dy - \log x - ...
3
votes
2answers
350 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})}} ...
3
votes
1answer
231 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 ...
3
votes
1answer
618 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 ...
3
votes
1answer
130 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
0answers
64 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} ...
3
votes
0answers
236 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} ...
3
votes
0answers
162 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 ...
2
votes
1answer
180 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, ...
2
votes
2answers
71 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
129 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
80 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 ...
2
votes
2answers
1k views

Hadamard's Gamma function

Hi, I'm looking for a link to a derivation of some of the basic properties of Hadamard's Gamma function. For instance that it satisfies $H(x+1)=xH(x)+\frac{1}{\Gamma(1-x)}$ I've been looking on the ...
2
votes
0answers
137 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 ...
2
votes
0answers
94 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 ...
2
votes
0answers
281 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 ...
2
votes
0answers
231 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 ...
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} ...
2
votes
1answer
627 views

What are conditions to make f(x) defined by f(x)=f(x-1)*x + 1/e unique(for instance convex)?

[Background:] Looking at the powerseries for the gamma-function $ \Gamma(1+x) = 1 + a_1 x + a_2 x^2 - a_3 * x^3 + ... $ then we can arrive at a decomposition $ \Gamma(1+x) = r(x) + g(x) $ ...