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
53 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
35 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
74 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 ...
0
votes
1answer
45 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
1answer
14 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
13 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
47 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
35 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
38 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
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0answers
28 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
132 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
15 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
188 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. $y=z^\frac12$ $=\frac12 e^{-z} z^ {1-\frac54} dz$ $=\Gamma\frac54$ ...
1
vote
1answer
70 views

Special Integral Proof

How to prove $$\int_0^\infty x^{2n-1} \exp(-a^{x^3})\, dx = \frac{\Gamma(n)}{2a^n} ,\quad n> 0 ,\quad a>0. $$
1
vote
1answer
15 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 $
1
vote
1answer
16 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 ...
1
vote
1answer
18 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
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0answers
39 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
35 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
44 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 ...
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0answers
22 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
84 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 ...
0
votes
0answers
25 views

Integrate $x^{7/3}e^{-x}$ from $0$ to $\infty$

Yes, there's the method of converting this function to gamma function of 10/3. Is there any other way?
4
votes
1answer
74 views

How to prove $H\left(\frac{1}{4}\right)=\frac{e^{\frac{C}{4\pi}-\frac{3}{32}}\cdot A^{\frac{9}{8}}}{\sqrt 2}$

How to prove: $$ H\left(\frac{1}{4}\right)=\frac{e^{\frac{C}{4\pi}-\frac{3}{32}}\cdot A^{\frac{9}{8}}}{\sqrt 2} $$ Where $C$ is Catalan's number, $A$ is Glaisher-Kinkelin's constant and $H(x)$ is the ...
7
votes
3answers
135 views

Log Gamma integral

What is the constant $\phi$ in the evaluation \begin{align} \int_{0}^{1/4} \ln\Gamma\left( t + \frac{1}{4}\right) \, dt = \frac{1}{8} \ln(\phi) \end{align} and the constant $\theta$ in the value ...
4
votes
0answers
53 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 = ...
0
votes
1answer
23 views

asymptotic series for “stable distribution”

I'm trying to understand how to get from one equation to another in a certain paper I am studying (DOI:10.1080/00018738100101467, eqs. 4.34 and 4.35). The equations are pretty self contained, so I'm ...
2
votes
2answers
59 views

Proving $z\,\Gamma(z) = \Gamma(z+1)$ using the product formula

I am trying to show that $z\,\Gamma(z) = \Gamma(z+1)$ using the product formula: $$ \Gamma(z) = \frac{e^{-\gamma z}}{z} \prod\limits_{n=1}^\infty\left(1+\frac{z}{n}\right)^{-1}e^{z/n}$$ where $\gamma$ ...
2
votes
0answers
19 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 ...
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0answers
38 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. $a_1=1$, ...
0
votes
1answer
67 views

ratio distribution of gamma with different parameter

Let $X$ be gamma distributed random variable with parameters $a$ and $b$. Let $W$ be gamma distributed random variable with parameters $c$ and $d$, such that \begin{equation} f_X(x) = ...
0
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2answers
59 views

Let$\ x$ be a real number between$\ 0$ and$\ 1$. Is it possible to write$\ e^{x}$ as a function of$\ \Gamma \left(x+1\right)$?

In particular, I'm looking for a relation between$\ e^x$ and$\ \frac{1}{ \Gamma \left(x+1\right) }$, which would be of help for a proof.
0
votes
1answer
59 views

Log integrals III

The integral \begin{align} J_{m} = \int_{0}^{1} \frac{t^{m}}{1+t} \, \ln(1+t) \, dt \end{align} has the general form \begin{align} J_{m} = (-1)^{m} \left[ A_{m} - B_{m} \, \ln(2) + C_{m} \, ...
2
votes
1answer
37 views

Prove a simplification of $B_{\frac{1}{2}}(n,n+1)$ for all complex n

$$B_{\frac{1}{2}}(n,n+1)=\frac{2^{-2 n-1} \left(\Gamma \left(n+\frac{1}{2}\right)+\sqrt{\pi } n \Gamma (n)\right)}{n \Gamma \left(n+\frac{1}{2}\right)}$$ for $n>0$ where $B$ is the incomplete beta ...
7
votes
2answers
149 views

A Sine integral: problem I

Is it possible to demonstrate a solution for the integral \begin{align} \int_{0}^{\infty} x^{n} \, \sin\left( a x^{2} + \frac{b}{x^{2}} \right) \, dx \end{align}
1
vote
2answers
124 views

Functional equation involving gamma function

Recently, I found the following functional equation: $$ n^{nx-1}\cdot\prod_{k=0}^{n-1}{\Gamma{\left(x+\frac{k}{n}\right)}}=\Gamma{(nx)}\cdot\prod_{k=1}^{n-1}{\Gamma{\left(\frac{k}{n}\right)}} $$ Now ...
8
votes
3answers
226 views

Evaluate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$

I have no idea where to even start. WolframAlpha cant compute it either. $$\int^1_0 \log^2(1-x) \log^2(x) \, dx$$ I think it can be done with series, but I am not sure, can someone help a little so ...
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0answers
18 views

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

This question has not got any answer on Mathoverflow. I admit that it is unusual to cross-post in this direction (from MO to math.SE), but knowing that some of those really unbelievable integrals tend ...
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0answers
55 views

Infinite product: Problem I

What is the closed form value of the infinite product $$ ...
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vote
2answers
18 views

conditional distribution of $A\mid N = k$

Can someone help me with how to get part 3 of the question?
3
votes
0answers
50 views

Confused about pochhammer contour?

I know some theorems about complex analysis such as the argument principle. But I do not get the pochhammer contour. I read about it on the wiki page of the beta function , but I do not understand a ...
0
votes
2answers
64 views

Approximating the Digamma fucntion near 1

Peace be upon you, I had the following system of equations to be solved \begin{align*} \begin{cases} \psi(\alpha)-\psi(\alpha+\beta)=c_1\\ \psi(\beta)-\psi(\alpha+\beta)=c_2 \end{cases} \end{align*} ...
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0answers
44 views

Liouville's Extension of Dirichlet Theorem

Can we use Liouville's Extension of Dirichlet Theorem to find triple integral $\int\int\int(u^2+v^2+w^2)\space du\space dv\space dw\space where\space u=0, v=0, w=0\space \&\space u+v+w\leq1$? Or ...
1
vote
0answers
20 views

Bounds on functions involving CDFs or Beta function

I have functions of the form \begin{align} I_i = \int_0^\infty F_0(x)^aF_1(x)^b(1-F_0(x))^c(1-F_1(x))^ddF_i(x)~~~~i = 0,1 \end{align} $F_0(x)$ and $F_1(x)$ are CDFs corresponding to the random ...
0
votes
0answers
64 views

Joint density function of exponential and gamma distribution

My problem is: $X_1,...,X_n$ are independent exponentially distributed random variables with $\lambda=1$ paremeters. I have to find the joint density funcitions of $ Y=\sum\limits_{i=1}^n{X_i}$ ...
0
votes
1answer
67 views

Integral using Beta Function and Gamma Function

Interestingly, I seem to have an integral I have posted before, but I want to take a different approach to it. $\int_{0}^{1} \frac{\ln(1+x)}{1+x^2} \,dx$ The beta function states, $B(x,y) = ...
1
vote
0answers
22 views

Integral with a gamma functions inside

I have a function based on the binomial distribution, $$f(x;n,p)=\sum_{i=0}^{n} |x-i|\binom{n}{i} p^i (1-p)^{n-i}.$$ It's not so hard to plot this out with discrete points, but I'd like to smooth ...
1
vote
2answers
28 views

Approximation involving Gamma function: $\frac{\Gamma(j+d)}{\Gamma(j+1)\Gamma(d)}\approx(j-1)^{d-1}$

With $d\leq 1$ and $$ a_j=\frac{\Gamma(j+d)}{\Gamma(j+1)\Gamma(d)}=\frac{d(d+1)\ldots(d+j-1)}{j!},\quad j=0,1,2,\ldots $$ my professor wrote in class that $$ \sum_{j=N}^\infty ...