3
votes
0answers
29 views

Closed form of $\int_0^1 \left(\ln \Gamma(x)\right)^3\,dx$

From the amazing result by Raabe we know that $$LG_1=\int_0^1 \ln \Gamma(x)\,dx = \frac{1}{2}\ln(2\pi) = -\zeta'(0).$$ We also know that $$LG_2 = \int_0^1 \left(\ln \Gamma(x)\right)^2\,dx = ...
2
votes
0answers
14 views

Do the incomplete gamma functions have reflection formulas?

Euler gave this reflection formula for the gamma function: $$\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin(\pi z)}$$ My question - do the lower incomplete gamma function $\gamma(s,x)$ and the upper ...
1
vote
0answers
32 views

Non-recursive closed-form of the coefficients of Taylor series of the reciprocal gamma function

The reciprocal gamma function has the following Taylor series. $$\frac{1}{\Gamma(z)}=\sum_{k=1}^{\infty}a_kz^k,$$ where the $a_k$ coefficient are given by the followint recursion. $a_1=1$, ...
3
votes
1answer
181 views

On definition of gamma function.

We all know that gamma function's definition is $$\Gamma \left( x \right) = \int\limits_0^\infty {s^{x - 1} e^{ - s} ds}$$ and it is divergent for $x<0$. Yesterday, I was studying about Bessel ...
0
votes
1answer
28 views

Closed form for this incomplete gamma series?

The series I'm working with is $$\sum_{k=0}^\infty \binom{z}{k}(-1)^k ( 1-\frac{\Gamma(k,-\log n)}{\Gamma(k)} )$$ with $z$ a complex variable and $\Gamma(k, -\log n)$ the upper incomplete gamma ...
1
vote
2answers
40 views

Series expansion of incomplete gamma function ratio

I am interested in the series expansion of: $$S(k)=\frac{\Gamma(k+1,a)}{k!},$$ around $k=\infty$ where $\Gamma(x,z)$ is the incomplete gamma function and $a$ is some positive constant. In ...
6
votes
3answers
119 views

A closed-form of product the gamma functions containing $\pi$ and $\phi$

Playing with gamma functions by randomly inputting numbers to Wolfram Alpha, I got the following beautiful result \begin{equation} ...
13
votes
1answer
254 views

A Gamma limit $\lim_{n\rightarrow+\infty}\sum_{k=1}^n \displaystyle \left( \Gamma\bigl(\frac{k}{n}\bigr)\right)^{-k}=\frac{e^\gamma}{e^\gamma-1}$

Show that $$\lim_{n\rightarrow+\infty}\sum_{k=1}^n \displaystyle \left( \Gamma\bigl(\frac{k}{n}\bigr)\right)^{-k}=\frac{e^\gamma}{e^\gamma-1}$$ where $\gamma$ is the Euler-Mascheroni Constant. ...
11
votes
2answers
115 views

Trigamma identity $4\,\psi_1\!\left(\frac15\right)+\psi_1\!\left(\frac25\right)-\psi_1\!\left(\frac1{10}\right)=\frac{4\pi^2}{\phi\,\sqrt5}.$

I heuristically discovered the following identity for the trigamma function, that I could not find in any tables or papers or infer from existing formulae (e.g. [1], [2], [3], [4], [5], [6]): ...
2
votes
1answer
36 views

Computing of the Gamma Function

I have stumbled upon Gamma functions when dealing with Gamma distributions on my studies with basic statistics. However, I have not understood how its computation expands factorials to real and ...
7
votes
1answer
70 views

A generalization of Bell numbers to arbitrary complex arguments

For $n\in\mathbb N$, the Bell number $B_n$ is a number of ways to partition the integer range $[1,\,n]$ into pairwise disjoint non-empty subsets. E.g. $B_3=5$ because ...
0
votes
0answers
69 views

Solving integral that contain upper incomplete gamma function, exponential, and powers

I have this integration formula; $ f=\int\limits_{0}^{\infty}\frac{e^{-b~z}}{\sqrt{z}} \Big(\frac{\beta}{\beta+z}\Big)\Big(\frac{\beta+z}{z}\Big)^L ...
3
votes
1answer
56 views

Is there a simpler form for $\Re \frac{\Gamma(1/2-i)}{\Gamma(1-i)}$?

Is there a simpler (i.e. manifestly real) form for $\Re \frac{\Gamma(1/2-i)}{\Gamma(1-i)}$ or $\Im \frac{\Gamma(1/2-i)}{\Gamma(1-i)}$, or more generally for $\frac{\Gamma(1/2-ia)}{\Gamma(1-ia)}$ with ...
1
vote
2answers
60 views

Gamma Function Result

I want to know the way to prove that $$ \Gamma(n + 1/2)= \frac{(2n)! \sqrt{\pi}}{4^n n!}. $$ I tried writing term by term and it gives the result, but I want to know how to prove it without ...
7
votes
4answers
195 views

Integral $\int_0^1 \log \left(\Gamma\left(x+\alpha\right)\right)\,{\rm d}x=\frac{\log\left( 2 \pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha$

Hi I am trying to prove$$ I:=\int_0^1 \log\left(\,\Gamma\left(x+\alpha\right)\,\right)\,{\rm d}x =\frac{\log\left(2\pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha\,,\qquad \alpha \geq 0. $$ I am ...
5
votes
2answers
193 views

Integral $\int_0^1 \log \Gamma(x)\cos (2\pi n x)\, dx=\frac{1}{4n}$

$$ I:=\int_0^1 \log \Gamma(x)\cos (2\pi n x)\, dx=\frac{1}{4n}. $$ Thank you. The Gamma function is given by $\Gamma(n)=(n-1)!$ and its integral representation is $$ \Gamma(x)=\int_0^\infty t^{x-1} ...
0
votes
1answer
40 views

factorization of an expression involving gamma function

Does the equation $\Gamma(x+1/2)\Gamma(x-1/2)=\Gamma(x+iy)\Gamma(x-iy)$, where $\Gamma(z)$ is the Gamma function and $i=\sqrt{-1}$, have any solution assuming $x,y$ are both real and $x>1/2$? This ...
2
votes
1answer
108 views

Gamma Function Contour Integration

So, I've been trying to prove the following integral related to the gamma function, and I'm really banging my head against the wall over this: ...
3
votes
0answers
87 views

integral involving incomplete gamma function

Need to evaluate the integral $$ \int_a^b e^{1/x}\,\Gamma(m,1/x)\,dx $$ or equivalently $$ \int_{1/a}^{1/b} y^{-2}\,e^{y}\,\Gamma(m,y)\,dy, $$ where $m$ is an integer, and $0<a<b<\infty$. The ...
2
votes
2answers
58 views

Integration by using special functions

$$\int ^{\pi }_{0}\dfrac {dt}{\sqrt {3-\cos t}}$$ How can you solve the following equation by using alpha/gamma functions and putting $$\cos t=1-2\sqrt {u}$$
0
votes
1answer
294 views

Integral of incomplete gamma function

I am trying to integrate this: $$\int_0^\infty z^{-|M|-1}\,\Gamma(A,z)\;dz$$ where $A$ is a real positive, and note that the power of $z$ is $-|M|-1$, i.e., is forced to be negative real.
0
votes
0answers
74 views

Learning about the gamma function.

I have just started learning about the gamma function but the books I have are not sufficient to give me a complete picture of it. Can you guys suggest some online resources/free books where I can ...
11
votes
1answer
235 views

Strange closed forms for hypergeometric functions

So in the process of trying to find a derivation for this answer, the following interesting equalities arose (one can check with Wolfram Alpha/Mathematica): $$\frac{8\sqrt{2}G^4}{5\pi^2} ...
0
votes
2answers
64 views

Expressing $\mathrm{B}(\sinh(x), \cosh(x))$ in terms of elementary functions

Is it possible to express: $\mathrm{B}(\sinh(x), \cosh(x))$ (where $\mathrm{B}$ is the beta function) In closed form, in terms of elementary functions?
0
votes
0answers
19 views

$\lim _{N\to \infty } Q^{-1}(\frac{N}{2},0.5)= ? $

What is $\lim _{N\to \infty } Q^{-1}(\frac{N}{2},0.5)= ? $ for $N>0$ where: $Q^{-1}(s,z)$ is the "Inverse of regularized incomplete gamma function" for $x$ in $z=Q(s,x)$. ...
1
vote
0answers
55 views

Asymptotic behavior of the Beta function

Let $B(z_1,z_2)$ be the Beta function, $z_1 = x_1 + iy_1$, $z_2 = x_2 + i y_2$. Suppose that $x_1$, $x_2 > 0$. I want to estimate the behavior of $|B(x_1+iy_1,x_2+iy_2)|$ as $|y_1|+|y_2|\to \infty$ ...
15
votes
2answers
328 views

Continuous generalization of $\sum_{k=0}^n {n \choose k} = 2^n$?

We know that $$\sum_{k=0}^n {n \choose k} = 2^n.$$ A continuous generalization of the formula would be $$\int_0^{n+1} \frac{\Gamma(n+1)}{\Gamma(n-x+1) \Gamma(x+1)} dx = 2^n?,$$ but this is incorrect ...
3
votes
2answers
171 views

Two series involving the Gamma function

The last piece I am left with in my proof is to compute the following two series: $$\sum_{i=1}^{n-1}\dfrac{\Gamma(i-d)\Gamma(n-i+d)}{\Gamma(i+1)\Gamma(n-i)(n-i-d)}, \sum_{i=1}^{n-1} ...
9
votes
2answers
280 views

The most complete reference for identities and special values for polylogarithm and polygamma functions

I am looking for a book, paper, web site, etc. (or several ones) containing the most complete list of identities and special values for the polylogarithm $\operatorname{Li}_s(z)$ and polygamma ...
1
vote
2answers
218 views

Derivation of Wallis's Formula

Use Euler's product formula $\Gamma (z)={1\over z}\prod_{n=1}^\infty({1+{1\over n}})^z({1+{z\over n}})^{-1}$ the fact that $\Gamma ({1\over 2})=\sqrt\pi$ to prove Wallis's Formula ${\pi\over ...
3
votes
1answer
104 views

Interesting integral involving $\Gamma (z)$.

Find the value of $$\int_0^\infty t^{x-1}e^{-\lambda t \cos(\theta)} \cos(\lambda t \sin (\theta)) dt$$ where $\lambda >0$, $x>0$, and ${-1\over 2}\pi < \theta < {1\over 2}\pi$ in terms of ...
4
votes
1answer
140 views

Inequality $\Gamma(\alpha+x)\Gamma(\beta+y)\leq C(\alpha; \beta)\Gamma(x+y-1)$

I need to prove $$\Gamma(\alpha+x)\Gamma(\beta+y)\leq C(\alpha; \beta)\Gamma(x+y),$$ for $x>\alpha$ and $y>\beta$ with $0<\alpha,\beta \leq \frac{1}{2}$ constants, and $C(\alpha, \beta)$ is a ...
1
vote
1answer
855 views

Relation between Beta and Gamma distributions

How do I demonstrate the relationship between the Beta and the Gamma function, in the cleanest way possible? I am thinking one (or two) substitution of variables is necessary, but when and how is the ...
4
votes
1answer
193 views

The infinite product representation of the Barnes G function

I want to show that the infinite product representation of the Barnes G function, namely $$ G(z+1)=(2\pi)^{z/2}\text{exp}\left(-\frac{z(z+1)}{2}- \frac{\gamma z^{2}}{2}\right)\, ...
23
votes
2answers
519 views

Evaluating $\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} $

Wolfram MathWorld states that $$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = \frac{ \pi \sqrt{3}}{18} \Big[ \psi_{1} \left(\frac{1}{3} \right) - \psi_{1} \left(\frac{2}{3} \right) \Big]- ...
2
votes
0answers
117 views

Limiting behavior of an integral involving incomplete Gamma function

I am wondering about the limiting behavior as $k\rightarrow\infty$ of the following integral: $$I(k)=\frac{2^{-k/2}}{\Gamma(k/2)}\int_{f(k)}^\infty ...
2
votes
2answers
91 views

Identity involving the rising factorial

I am reading a book about hypergeometric functions and in a proof of a transformation they use the supposedly obvious fact $$ \displaystyle\frac{(c-a-b)_{n-r}}{(n-r)!} = \frac{(c-a-b)_{n} ...
5
votes
0answers
244 views

Inequality between incomplete beta and gamma functions

Let the regularized incomplete beta and gamma functions be defined as usual: \begin{equation} I_p(z,w) = \frac a {B(z,w)} \int_0^p t^{z-1} (1-t)^{w-1} \,\mathrm dt, \end{equation} \begin{equation} ...
10
votes
2answers
304 views

The Laurent series of the digamma function at the negative integers

To find the Laurent series of $\psi(z)$ at $z= 0$, I would first find the Taylor series of $\psi(z+1)$ at $z=0$ and then use the functional equation of the digamma function. Specifically, ...
1
vote
2answers
83 views

Why do we have $\psi^{(m)} (z)=(-1)^{m+1}\int_{0}^{\infty}\frac{t^me^{-zt}}{1-e^{-t}}dt$?

Why is the following representation true? $$\psi^{(m)} (z)=(-1)^{m+1}\int_{0}^{\infty}\frac{t^me^{-zt}}{1-e^{-t}}dt,$$ where $\psi^{(m)} (z)$ denotes the Polygamma function.
3
votes
1answer
550 views

Real and imaginary part of Gamma function

Is there a way to separate the real and imgainary part of the gamma function $$\Gamma (a+ib)$$ I thought of using the formula $$\zeta(z) \Gamma(z) = \int^{\infty}_0\frac{t^{z-1}}{e^t-1}\, dt$$ ...
0
votes
1answer
43 views

Non-equality with Gamma functions

Let $n \in N$, $k \in Z_+$. Show that $$ \frac{\Gamma\left(k+\frac 12\right)}{\Gamma\left(k+\frac ...
3
votes
2answers
82 views

Upper bound for the quotient of gamma functions?

I am looking for an upper bound for $$ \frac{\Gamma(x+\beta)}{\Gamma(x)},\,\,\,\beta>0.$$ In this question it was shown that $$ \frac{\Gamma(x+\beta)}{\Gamma(x)} \approx x^\beta. $$ Then, I ...
13
votes
7answers
668 views

Evaluating $\int_0^\infty \frac{dx}{1+x^4}$. [duplicate]

Can anyone give me a hint to evaluate this integral? $$\int_0^\infty \frac{dx}{1+x^4}$$ I know it will involve the gamma function, but how?
5
votes
1answer
153 views

Integration gamma and beta: $\int_0^4y^3\sqrt{64-y^3}\,\mathrm dy$

How can we evaluate the following integral? $$\int_0^4y^3\sqrt{64-y^3}\,\mathrm dy$$ I can't find anything to substitute because all of the trigonometric identities are in square form...
4
votes
1answer
254 views

Sum of Infinite Series with the Gamma Function

I am computing the volume of an infinite family of polytopes and have run into the following sum, which I am not sure how to evaluate, as it looks similar to the Riemann zeta function, except with the ...
4
votes
1answer
136 views

Integral formula for $\frac{1}{\Gamma(z)}$

Let $c>0$. How to prove that for any complex number $z$, $$\frac{1}{\Gamma(z)}=\frac{1}{2\pi}\int_{-\infty}^\infty (c+it)^{-z}e^{c+it}\,dt?$$ where $\Gamma(z)$ is the Gamma function.
37
votes
1answer
1k views

Is it possible to simplify $\frac{\Gamma\left(\frac{1}{10}\right)}{\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$?

Is it possible to simplify this expression? $$\frac{\displaystyle\Gamma\left(\frac{1}{10}\right)}{\displaystyle\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$$ Is there a systematic ...
3
votes
2answers
328 views

Evaluate integral in terms of Gamma function

Need help evaluating the following integral in terms of the gamma function where gamma function is: $$\Gamma(x)=\int_0^\infty e^{-t}t^{x-1}dt.$$ The integral is the following: $$\large{\int_0^\infty ...
17
votes
1answer
238 views

Is $K\left(\frac{\sqrt{2-\sqrt3}}2\right)\stackrel?=\frac{\Gamma\left(\frac16\right)\Gamma\left(\frac13\right)}{4\ \sqrt[4]3\ \sqrt\pi}$

Working on this conjecture, I found its corollary, which is also supported by numeric caclulations up to at least $10^5$ decimal digits: ...