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

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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}) ...
2
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0answers
24 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 ...
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0answers
20 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
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1answer
48 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 ...
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0answers
21 views

CDF of ratio of Gamma distribution with different parameters

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) = ...
3
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2answers
69 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 ...
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0answers
30 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 = ...
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1answer
20 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 ...
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2answers
43 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 $\gamma$ ...
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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 ...
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0answers
35 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
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1answer
60 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) = ...
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2answers
58 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.
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1answer
54 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} \, ...
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1answer
34 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
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2answers
141 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}
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2answers
122 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 ...
6
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3answers
153 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
12 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
51 views

Infinite product: Problem I

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

conditional distribution of $A\mid N = k$

Can someone help me with how to get part 3 of the question?
2
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0answers
39 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 ...
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0answers
32 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 ...
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0answers
17 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 ...
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0answers
29 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
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1answer
50 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) = ...
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0answers
21 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 ...
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2answers
27 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 ...
10
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2answers
146 views

Computing a limit involving Gammaharmonic series

It's a well-known fact that $$\lim_{n\to\infty} (H_n-\log(n))=\gamma$$ Now, if I change things a bit and use the fact that $\displaystyle \Gamma \left( \displaystyle \frac{1}{ n}\right) \approx n$ ...
1
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1answer
21 views

Derivatves of curves of hyper-sphere volumes and areas

See wikipedia "N-sphere". I need this differentiated with respect to "n", not "r". This is so I can find when the slope of the curve with respect to n(treated as x-axis), the number of dimensions, ...
2
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2answers
54 views

Is the gamma function $\Gamma(n+1)$ the only continuous function and defined derivative with the same recursive definition as of $n!$ for $n>-1$? [duplicate]

Is the gamma function $\Gamma(n+1)$ the only continuous function and defined derivative with the same recursive definition as of $n!$ for $n>-1$ ? (When using real numbers.) The recursive ...
1
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1answer
41 views

Series involving a Logarithm

Consider the series \begin{align} \sum_{n=1}^{\infty} \left[ \frac{n}{a} \ln\left(1 + \frac{a}{n}\right) - 1 + \frac{a}{2n} \right]. \end{align} Is there a closed form solution to this series and what ...
0
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1answer
35 views

Bounds on the real and imaginary parts of the digamma function $\psi $

Let $\psi $ be the digamma function given by $$\psi (z)=\left.\frac {d}{dt}\log\Gamma (t)\right|_{t=z}. $$ I wonder does anyone know of any lower and/or upper bounds on the real and imaginary parts ...
3
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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 ...
6
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1answer
177 views

Closed form for $\int_1^\infty\frac{\operatorname dx}{\operatorname \Gamma(x)}$

Is a closed form for $$\int\limits_1^{+\infty}\frac{\operatorname dx}{\operatorname \Gamma(x)}$$known? I tried to find it, but all well-known integrals involving gamma-function (such as of ...
0
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2answers
44 views

Ratio of Gamma random variables

If $X_i$, $i=1,2$ are independent gamma$(\alpha_i,1)$ random variables, find the distribution of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$. Attempt: Let $Y_1 = \frac{X_1}{X_1+X_2}$ and ...
0
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0answers
13 views

gamma function implementation for negative x

I am using MATLAB to calculate incomplete gamma function. I would like to ask that if $x$ is negative, how can we assure that the value is calculated correctly?
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0answers
12 views

Deriving the multivariate t-distribution from the normal mixture representation

I'm trying to derive multivariate t-distribution from its representation as a normal variance mixture distribution by following the calculations in Appendix 4 of ...
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2answers
89 views

Compute $\Gamma(2.7)$

All I know of the Gamma function is $$\Gamma(\alpha) = \int\limits_{0}^{\infty}x^{\alpha - 1}e^{-x}\text{ d}x$$ and the recursive formula $$\Gamma(\alpha) = (\alpha-1)\Gamma(\alpha - 1)\text{.}$$ A ...
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1answer
20 views

Exponential random variable with mean 1/gamma

If $X$ is an exponential random variable with mean $\frac{1}{γ}$, show that $\mathbb{E}[X^k]=\frac{k!}{γ^k},\,\, k=1,2,3,\cdots$ *Use the gamma density function $\mathbb{E}[X^k]=∫x^{k}γe^{-γx}dx$ ...
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2answers
67 views

Why this gamma function reduces to the factorial?

$$\Gamma(m+1) = \frac{1\cdot2^m}{1+m}\frac{2^{1-m}\cdot3^m}{2+m}\frac{3^{1-m}\cdot4^m}{3+m}\frac{4^{1-m}\cdot5^m}{4+m}\cdots$$ My books says that in a letter from Euler to Goldbach, this expression ...
0
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1answer
28 views

Arc-Gamma Function.

Is there an arc-gamma function? Where gamma(x) = y... Arc-gamma(y) = x. I've searched and found something called DiGamma Function, but when I substituted it didn't seem to be "arc" but something ...
1
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1answer
31 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 ...
7
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2answers
107 views

Why is this function a really good asymptotic for $\exp(x)\sqrt{x}$

$$f(x)=\sum_{n=0}^{\infty} a_n x^n\;\;\;\;\; a_n = \frac{1}{\Gamma(n+0.5)}$$ Why is this entire function a really good asymptotic for $\exp(x)\sqrt{x}$, where for large positive numbers, ...
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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 ...
3
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0answers
40 views

An integral similar to integrating the Beta function

Peace be upon you, I have the following definite integral for the mathematical expectation of some distribution \begin{align*} \int_{-1}^1 z^2\int_{\max(z-1,-1)}^{\min(0,z)} (z-y)^a(-y)^b \ ,dy \,dz ...
1
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1answer
34 views

Generalized series expansion for $\Gamma(a,z)$ in $a$ at $a=0$

I need the generalized series expansion of $\Gamma(a,z)$ for $z\in\mathbb{C}$ and $a\to0^+$. Mathematica gives a result that seems to be correct, but I have to make sure of its validity. I came ...
0
votes
1answer
23 views

A converse theorem for Hecke modular forms?

Let $f$ be a function which can be represented by a Dirichlet serie (for $\Re s$ big enough) $$f(s)=\sum_{n=1}^{\infty}{\frac{a(n)}{n^s}}$$ and which can be meromorphically extend to $\mathbb{C}$ by ...
7
votes
0answers
130 views

Is it true that $\gamma_{\lfloor\log\Gamma(x)\rfloor}\sim 2\pi x$?

I realise that Gram points can approximate the imaginary part on the $x$th zeta zero $(\gamma_x)$ accurately, and indeed, Guilherme França, André LeClair give another formula, namely ...
2
votes
2answers
60 views

Factorial as a sum. Insight appreciated

I recently posted an answer to a question about ways to express the factorial function as a sum. I posted the following formula, which I discovered several years ago and I haven't seen anywhere else: ...