The gamma-function tag has no wiki summary.

**66**

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**9**answers

7k 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 ...

**42**

votes

**6**answers

2k 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:
...

**13**

votes

**1**answer

316 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 ...

**12**

votes

**2**answers

1k 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 ...

**12**

votes

**2**answers

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 ...

**11**

votes

**1**answer

245 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 ...

**9**

votes

**0**answers

476 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 ...

**8**

votes

**3**answers

804 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 ...

**7**

votes

**5**answers

730 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 ...

**7**

votes

**1**answer

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) = ...

**6**

votes

**1**answer

187 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

**1**answer

266 views

### Inequality for a gamma function

Let $s=\sigma+it$ and $\Gamma(s)$ be the Euler gamma function.
Does the inequality holds?
$$
\left|\frac{\Gamma(s)}{\Gamma(2-s)}\right|\leq |s|^{2(\sigma-1)},\, 1<\sigma<2,\, t>t_0>0.
$$
...

**5**

votes

**4**answers

502 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

**5**answers

721 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

**2**answers

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 ...

**5**

votes

**1**answer

226 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 ...

**4**

votes

**1**answer

585 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

**2**answers

534 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

**1**answer

565 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 ...

**4**

votes

**1**answer

416 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} ...

**3**

votes

**3**answers

471 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

**0**answers

216 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

**0**answers

145 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

**2**answers

290 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})}}
...

**2**

votes

**1**answer

131 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

**1**answer

549 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 ...

**2**

votes

**2**answers

775 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

**0**answers

166 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**

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**0**answers

297 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 ...

**2**

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**0**answers

193 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

**0**answers

512 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

**1**answer

512 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) $
...

**2**

votes

**0**answers

340 views

### 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 ...

**1**

vote

**3**answers

473 views

### Undefined gamma function problem

Hello,
I'm trying to solve the following integral :
$\int_0^\infty \frac{1}{t^{d/2}}(e^{-\gamma t} - e^{-\delta t})dt$.
I know it equals
...

**1**

vote

**0**answers

194 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 ...

**1**

vote

**0**answers

244 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 $\epsilon \rightarrow 0$,
...

**1**

vote

**0**answers

198 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 ...

**1**

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**0**answers

176 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 ...

**0**

votes

**3**answers

250 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}) = ...

**0**

votes

**1**answer

775 views

### Generalizations of a product formula for the gamma function

Hello and Happy holidays.
I am interested in generalizations of the following product formula for the gamma function
$\Gamma(z)= \int_{0}^{\infty} t^{z-1}e^{-t}dt$ when $n \geq 2$:
\begin{align}
...

**0**

votes

**1**answer

497 views

### Calculate conditional expectation of log(x) with gamma density

How to calculate the following expression:
$$\int_0^u{\ln(x)x^{k-1}e^{-x}}\;dx$$
As I know,
$$\int_0^\infty{\ln(x)x^{k-1}e^{-x}}dx = \Gamma(k)\Psi(k)$$
Are there any way to transfer the integral ...

**0**

votes

**0**answers

64 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 ...

**0**

votes

**1**answer

439 views

### maximum likelihood of gamma distribution computer calculation

My problem is that given a dataset, I want to program fitting a gamma distribution on this data by estimating the two parameters(shape and the scale parameters) using Maximum Likelihood Estimation. I ...