Questions on the gamma function $\Gamma(z)$ of Euler extending the usual factorial $n!$ for arbitrary argument, and related functions.

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5
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1answer
77 views

Multiple integrals involving product of gamma functions

The following integral was posted a few days back on Integrals and Series forum: $$\int_0^{2\pi} \int_0^{2\pi} \int_0^{2\pi} \frac{dk_1\,dk_2\,dk_3}{1-\frac{1}{3}\left(\cos k_1+\cos k_2+ \cos ...
3
votes
0answers
65 views

What is $\Gamma(1+\Gamma(1+\Gamma(1+\dots)))$?

Consider the infinitely nested expression $$x=\Gamma(1+\Gamma(1+\Gamma(1+\dots)))$$ where $\Gamma$ is the Gamma function. Imitating the standard method for solving infinitely nested radicals, we ...
0
votes
1answer
38 views

how $\prod\limits_{i=1}^{n} (2k-1)/2= (2n)!/{(4^n)n!} = (2n-1)!/[{2^{2n-1}}(n-1)!]$? [on hold]

We know $Γ(n+1/2)=(n-1/2)!= Π (n-1/2)= √π \cdot \prod\limits_{i=1}^{n} (2k-1)/2$ hence $Γ(n+1/2) = (2n)!/{(4^n)n!}√π = (2n-1)!/[2^{2n-1}(n-1)!]\sqrtπ$ But I need the answer of above question to prove ...
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0answers
30 views

Sum of all n dimensional spheres?

I was messing around and made some code to find the area of an n dimensional sphere. I noticed that as n increases, the area tends towards zero. These were the results: ...
1
vote
1answer
30 views

Statistics question involving exponential distribution and (maybe) gamma function

This is from a past stat exam that I am studying for my final tomorrow (lol). I believe this might have to do with gamma function. Could someone guide me through step by step of how to do this? An ...
0
votes
1answer
22 views

Statistics questions (normal distribution and possibly gamma function)

This is a question from a past stat exam that I am studying because my final is in two days (lol). It'd be great if someone could guide me through how do both parts of the problem. I know gamma ...
1
vote
3answers
90 views

Value of $\sum_{n=0}^{\infty} \exp(-bn^a)$

What is the value of $$ S = \sum_{n=0}^{\infty} \exp(-bn^a) $$ where $a>0$ and $b>0$? I know that $$ \int_0^{\infty} \exp(-bx^a) \, \mathrm{d}x=\frac{1}{ab^{1/a}}\Gamma(1/a) $$ but cannot ...
1
vote
1answer
131 views

Help to solve complex equation related to the Gamma function

I would need some help to solve the next complex equation for $y\in\mathbb {R}$, which I already know to be real-valued: $$ \frac {1} {2i}\left ((2\pi)^{\text {iy}}\text {}\text {Sin}\left (\frac ...
1
vote
1answer
27 views

inequality regarding the gamma function

I am curious to know if the following is true: Let $\alpha >0$. Then \begin{equation} \int_0^K x^{\alpha -1} e^{-x} \,dx \leq \Gamma (\alpha), \quad \forall K \in \mathbb{R}. \end{equation} This ...
2
votes
3answers
77 views

Evaluating $\int_{0}^{1} \left ( \ln \frac{1}{x} \right)^ndx$.

I have found this result on the Internet $$\int_{0}^{1} \left ( \ln \frac{1}{x} \right)^ndx = \Gamma(n+1).$$ I know that if $n \in \mathbb{N}$, the proof is not complicated. However, if $n \in ...
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vote
0answers
18 views

Proving a bound for $\Gamma(s+u)/\Gamma(s)$

Suppose $s, w$ are complex numbers with positive real part. I have come across a particular bound that I have seen multiple times, but which I do not know how to prove: ...
1
vote
1answer
39 views

What non-integer number has the smallest factorial? [duplicate]

Quick google search for factorial of non-integers led me to gamma function. I tried that in my calculator and it worked as expected for non-integers. Perhaps implements gamma function internally. ...
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0answers
40 views

Is this proof show that the derivative of zeta function has no zeros in the critical strip?

suppose that :( $k \circ \zeta )(s)$= $(\zeta \circ k )(s) \neq 0 $.....$(1)$ ,and suppose that $ k(s)=\zeta(1-s)$ where : $\xi(s) = s(s-1) \pi^{s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s)$. and ...
1
vote
1answer
61 views

Zeta and Gamma function regularization with $\omega=1/0$

I have recently read about zeta function regularization, a way of ascribing values to functions having simple poles in a point and to divergent series. The values obtained are the same as those ...
3
votes
0answers
26 views

property of complex gamma function

Show that $$|(n+ib)!|= \left( \frac{\pi b}{\sinh(\pi b)}\right)^{{1}/{2}}\prod_{s=1}^{n}{(s^2+b^2)^{{1}/{2}}}$$ I have the following relations $$\frac{\sinh(b \pi)}{b\pi} = ...
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vote
0answers
31 views

How to compute this limit involving Gamma functions

I am investigating the following limit $$\lim_{u\to \infty}\frac{\Gamma \left(\frac{1}{2 H},\frac{H 2^{\left(\frac{1}{2H}\right)}}{u^{\frac{1}{H}}\left(1-H\right)^{\left(2 - \frac{1}{H}\right)}} ...
0
votes
0answers
16 views

Upper bound of $\int_{z_1}^{z_2}x^k e^{-x}dx$

How can I find a good upper bound to this quantity ? $$\int_{z_1}^{z_2}x^k e^{-x}dx,\ \ \ \ z_{1,2}\in \mathbb {Im}, k\in \mathbb N$$ The integrand makes me thinking about an incomplete gamma ...
3
votes
1answer
38 views

Closed form of the sequence $(3n)\times (3n-1) \times \cdots \times 6 \times 5 \times 3 \times 2$?

I need the closed form of the sequence $(3n)\times (3n-1) \times \cdots \times 6 \times 5 \times 3 \times 2$? The title is fairly self-explanatory. I know that a closed form of this exists using the ...
0
votes
2answers
32 views

Ratio of 2 Sums of products of binomial coefficients

I want to prove that for $k \ even, 0 \leq k<n, n\in \mathbb{N}$: $-\frac{1}{(2n-3-k)(k+2)}\sum \limits_{i=0}^{k} \frac{(-1)^{i} 2^{i} (2n-2-i)!}{(n-1-i)!i!(k-i)!}=\sum \limits_{i=0}^{k+2} ...
0
votes
3answers
70 views

Integral with $e$ , $ \int e^{x^3}\;dx$ [duplicate]

$ \int e^{x^3}\;dx$ so, i'm searching for answer this question, i think that this not is too easily, but i think this integral not exist solution undefined, this integral would be easy if had the ...
1
vote
1answer
24 views

Stirling formula on $\frac{\gamma(\frac{1}{2}s)}{\gamma(\frac{1-s}{2})} = O(|t|^{\sigma - \frac{1}{2}})$

I tried to prove $\frac{\Gamma(\frac{1}{2}s)}{\Gamma(\frac{1-s}{2})} = O(|t|^{\sigma - \frac{1}{2}})$ using stirling formula when $s =\sigma+it$. However, since stirling formular for gamma function is ...
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3answers
112 views

Evaluating this integral using the Gamma function

I was wondering if the following integral is able to be evaluated using the Gamma Function. $$\int_0^{\infty}t^{-\frac{1}{2}}\mathrm{exp}\left[-a\left(t+t^{-1}\right)\right]\,dt$$ I already have a ...
1
vote
1answer
29 views

Gamma of 3z using triplication formula:

I have to demostrate the gamma function for 3z as you see below: Using the multiplication formula demostrate gamma(3z) Gamma functions of argument $3z$ can be expressed using a triplication ...
3
votes
1answer
70 views

Solve this equation $\binom{4}{x}^2-\binom{4}{x}-25=0$

Can I solve this equation $$\binom{4}{x}^2-\binom{4}{x}-25=0$$
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0answers
42 views
1
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1answer
31 views

Log of 1-reguralized incomplete gamma function (upper)

I must compute value of 1-regularized incomplete gamma function (upper) $Q(a,z)$. But unfortunatelly this computing exceeding the precision of the processor (for example I gain 0, but I should have ...
5
votes
1answer
74 views

Pretty lower bound on the gamma function

According to http://functions.wolfram.com/06.05.29.0006.01, for every $x\geq 2$ it is $$ \left( \frac{x}{e}\right)^{x-1} \leq \Gamma(x) \leq \left( \frac{x}{2}\right)^{x-1}, $$ where $\Gamma$ is the ...
0
votes
2answers
36 views

Euler and the factorial function

I recently purchased H. M. Edwards' book entitled The Riemann Zeta Function. In the early pages of the volume, concerning the factorial function $\Gamma$, Edwards notes that "Euler observed that ...
7
votes
2answers
76 views

How does $\int_0^\infty e^{-t^4}dt = \Gamma (\frac{5}{4}) ?$

My text book claims that $$\int_0^\infty e^{-t^4}dt = \Gamma \left(\frac{5}{4}\right).$$ I fail to see this. By the definition of the gamma function we have $$\Gamma (z) = \int_0^\infty ...
1
vote
1answer
52 views

Integrate $\frac{\lambda y^{2}}{\sqrt{2\pi}} e^{-(\frac{1}{2}+ \lambda x)y^{2}}$ with respect to $y$

$$F(x,y)= \frac{\lambda y^{2}}{\sqrt{2\pi}} e^{-(\frac{1}{2}+ \lambda x)y^{2}}$$ Please show that the function when integrated with respect to $y$ is $F_X(x)= \frac{\lambda}{(2\sqrt{2}(\frac{1}{2}+ ...
3
votes
2answers
42 views

How does the recursion relation work in the solution to this differential equation (using series)?

Sorry for the vague title but it would not let me post the first step and last step of this equation (too many characters!). How does $$\dfrac{a_0}{3n(3n-1)(3n-3)(3n-4)\cdots 9 \cdot 8 \cdot 6 \cdot ...
0
votes
1answer
14 views

Gamma function identity used in deriving negative binomial from gamma-poisson mixture

On this wikipedia page negative binomial distribution, Negative binomial was derived as integrating out the lambda from Gamma-Poisson mixture. I tried to follow the proof step by step, but I am stuck ...
0
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1answer
50 views

How to prove this gamma function identity?

Reading Landau and Lifshitz "Quantum Mechanics. Non-relativistic theory", I've come across an identity, which after being a bit simplified, reads ...
5
votes
3answers
39 views

Find the summation $\sum_{k=0}^n (-1)^k \binom{n}{k}\frac{1}{s+k}$

Anyone can help me finding this summation: $$ \sum_{k=0}^n (-1)^k \binom{n}{k}\frac{1}{s+k}. $$ Where there is a similar one with known answer $$ \sum_{k=0}^n (-1)^k ...
0
votes
1answer
41 views

An integral involves Gamma function

Thanks for your attention, I meet an integral involves Gamma function and exponential function as follows:$$\int_a^\infty {{x^\alpha }} {e^{cx}}\Gamma \left( {s,bx} \right)dx$$ where $a > 0,s ...
0
votes
0answers
30 views

Solving an equation for $\lambda$

In trying to find an estimator for gamma data under type II censoring I was left with this equation to solve. I would like help in solving for $\lambda$. $$ \frac{mk}{\lambda} + ...
5
votes
2answers
142 views

Closed form for a zeta series

It is not that diffcult to derive \begin{align} \sum^\infty_{k=2}\frac{(-1)^{k-1}\zeta(k)}{k2^k}=&-\frac{\gamma}{2}+\ln\left(\frac{2}{\sqrt{\pi}}\right)\tag{1}\\ ...
0
votes
1answer
17 views

singular fnction involving gamma function

let be the function $$ \frac{\Gamma (m+1)}{\Gamma(m-2r+1)} $$ where m and r are integers... then my question is if for $ m<0 $ but $ r$ always positive integer the function above or its ...
13
votes
1answer
212 views

Integral that arises from the derivation of Kummer's Fourier expansion of $\ln{\Gamma(x)}$

I am trying to prove that for $0<x<1$, $$\color{blue}{\ln{\Gamma(x)}=\frac{1}{2}\ln(2\pi)+\sum^\infty_{n=1}\left\{\frac{1}{2n}\cos(2\pi nx)+\frac{\gamma+\ln(2\pi n)}{n\pi}\sin(2\pi ...
2
votes
1answer
50 views

Question about $\frac {\Gamma'(z+1)}{\Gamma(z+1)}$

If $\psi (z)= \log\Gamma(z+1)$ Prove that : $$\psi(n)+\gamma=1+\frac{1}{2}+\cdots+\frac{1}{n}$$ My Proof : $$\psi (z)= \frac {\Gamma'(z+1)}{\Gamma(z+1)}=-\frac{1}{z+1}-\gamma + \sum_{n=1}^\infty ...
1
vote
1answer
33 views

Prove that $\int_0^\infty y^{\frac12} e^{-y^2} \int_0^\infty y^{-\frac12} e^{-y^2} =\frac{\pi}{2^{\frac32}}$

My attempt: $\int_0^\infty y^{\frac12} e^{-y^2}$ $=\frac12 e^{-z} z^ {-\frac14} dz$, using the transformation: $y^2=z$, i.e. $y=z^\frac12$ $=\frac12 e^{-z} z^ {1-\frac54} dz$ $=\Gamma\frac54$ ...
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1answer
71 views

Special Integral Proof

How to prove $$\int_0^\infty x^{2n-1} \exp(-a^{x^3})\, dx = \frac{\Gamma(n)}{2a^n} ,\quad n> 0 ,\quad a>0. $$
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vote
1answer
16 views

derivatives of the inverse gamma function at negative integers??

what are the derivatives of the function $ \frac{1}{\Gamma(s)} $ at 0 and negative integers ?? i believe that $$ \frac{ d^{k}}{ds^{k}}\frac{1}{\Gamma(s)}=0 $$ for $ s=0,-1,-2,-3,-4 $
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vote
1answer
32 views

poisson gamma mixture

Let $N$ be Poisson distributed with parameter $10B$, where $B \sim \Gamma(3,1)$ (i.e. $f(b)=\frac{b^2 e^{-b}}{2}$). Find the p.m.f of $N$. How should I manipulate $10B$ in the integration? What is ...
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vote
1answer
20 views

Sum of complex digamma functions

It seems that the sum of the digamma function of $z$ and the digamma function of its conjugate $z^*$ is always real-valued. ...
2
votes
0answers
45 views

Ascribing values to Gamma of negative integers

It is widely known that $0^0$ is usually defined to be 1. I wonder why we cannot employ a similar technique to ascribe values to functions having poles in a point. Now take the Gamma function. The ...
2
votes
1answer
37 views

Integration of complex functions

Show that $\displaystyle \int_{-\infty}^\infty x^{2n}e^{-x^2}dx=(2n)!\frac{\sqrt\pi}{4^n n!}$ by differentiating the equation $\displaystyle \int_{-\infty}^\infty e^{-tx^2} dx=\sqrt{\frac{\pi}{t}}$. ...
1
vote
2answers
59 views

Integrating the gamma function

I assumed that $$\Gamma\left(k+\frac{1}{2}\right)=2\int^\infty_0 e^{-x^2}x^{2k}\,dx=\frac{\sqrt{\pi}(2k)!}{4^k k!} \,,\space k>-\frac{1}{2}$$ and that ...
0
votes
0answers
23 views

Simplifying Gamma functions yet having a complication while graphing when the function was able to be graphed previous to simplification?

According to the Euler's duplication formula: $$ \Gamma(z) \Gamma(z+\frac{1}{2}) = 2^{1-2z} \sqrt{\pi} \Gamma(2z) \therefore $$ $$ \Gamma(2z) = \frac{\Gamma(z) \Gamma(z+\frac{1}{2}) ...
7
votes
1answer
85 views

Integral involving hyperfactorial

I'm trying to prove that: $$ \int_0^1 \ln\left(K(x)\right)\space dx =-\zeta'(-1)=\ln(A)-\frac{1}{12} $$ Where $A$ is Glaisher Kinkelin's constant and $K(x)$ is a generalization of the hyperfactorial ...