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

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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}$ ...
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1answer
32 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
20 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
26 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 ...
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2answers
140 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 $\displaystyle \Gamma \left( \displaystyle \frac{1}{ n}\right) \approx n$ ...
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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, ...
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2answers
51 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 ...
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1answer
38 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 ...
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1answer
30 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 ...
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1answer
170 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 ...
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1answer
164 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 ...
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2answers
29 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 ...
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0answers
12 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
8 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
88 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
17 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
66 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 ...
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1answer
25 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 ...
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1answer
25 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 ...
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2answers
102 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
37 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 ...
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0answers
37 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 ...
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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 ...
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1answer
21 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
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0answers
127 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 ...
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2answers
39 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: ...
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0answers
13 views

Behavior of lower incomplete gamma function at complex infinity

The lower incomplete gamma function is given by $ \gamma\left(s, x\right) = \int\limits_0^x t^{s-1} e^{-t} {\rm d} t~,$ and has a well-defined analytic continuation for both $s$ and $x$ [1]. ...
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0answers
20 views

Showing distribution has a $\chi^2$ distribution with df = n

Let $X_1,X_2,....,X_n$ denote independent identically distributed random variables such that $X_1$ has density $p_1(x;\theta)$ where $\hspace{15mm}p(x;\theta) ...
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3answers
109 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} ...
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0answers
19 views

zeros of difference of incomplete Gamma functions

Gamma function is defined as: $$\Gamma(z):=\int_0^{\infty} t^{z} e^{-t} \frac{dt}{t}\tag{1}$$ The upper-incomplete Gamma function is defined as: $$\Gamma(z,a):=\int_a^{\infty} t^{z} e^{-t} ...
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0answers
18 views

Is it correct approximation of Upper Incomplete gamma function?

I am trying to approximate the Upper incomplete regularized gamma function $P(s,t)$, at the constant value of $s=c$ by: $$Q(c,t) \approx 1-Q(t,c)$$ where: $c$ is some constant and $t$ is variable. ...
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0answers
20 views

Stirling like approximation for lower-incomplete gamma function?

May we have a similar approximation for lower incomplete gamma function $\gamma(s,x)$, as we have a Stirling's approximation for Gamma function $\Gamma(s)$.
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4answers
77 views

The series $\sum_n\Gamma (n-1/3)/(n-1)!$ diverges

I would like to prove that the series: $$\sum_{n=1}^{\infty}\frac{\Gamma (n-1/3)}{(n-1)!}$$ diverges. The problem is that I don't know how to begin. Intuitively I get the result, because observing ...
5
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1answer
90 views

Expressing $\int_{-\infty}^\infty dx/(x^2+1)^n$ in terms of Gamma function

How to prove this identity for $n>1/2$? $$\int_{-\infty}^{\infty}\frac{dx}{(x^2+1)^n}=\frac{\sqrt{\pi}\cdot \Gamma(n-\frac{1}{2}) }{\Gamma (n)}$$
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1answer
27 views

Probability: Gamma Function vs Gamma Distribution

Could someone help me with setting up the function of this question. I've been setting it up with the gamma distribution function but kept getting the wrong answer. What I did was I used the Gamma ...
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2answers
97 views

Solving integral that contain exponential function and lower incomplete gamma function

I have the following integral; $$y=\int_0^\infty\frac{e^{-xf}}{m+x}\gamma(a,hx)~dx$$ where $f,m,h\in\mathbb{R}^+$ , $a\in\mathbb{N}$ , $\gamma\left(a,h x\right)$ is the lower incomplete gamma ...
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1answer
232 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. ...
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2answers
100 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]): ...
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0answers
55 views

Analytic solution for system of Digamma function equations

Peace be upon you, There is a while, I am seeking for the analytic solution of $\alpha$ and $\beta$ in this system \begin{align*} \begin{cases} \psi(\alpha)-\psi(\alpha+\beta)=c_1\\ ...
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1answer
33 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
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1answer
91 views

Fixed points of the Gamma Function?

I am interested in complex values of $z$ such that $$ \Gamma (z) =z$$ Clearly, the one trivial value of $z$ is 1. Also, looking at a graph of the gamma function on the real axis, I can tell that there ...
2
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1answer
68 views

Asymptotics of coefficients $[x^n] \frac{1}{\Gamma(1+x)}$ as $n$ is great

I am interested in the behaviour, as $n$ is great, of the coefficients $g_n$ in the Maclauren expansion of $\displaystyle \frac{1}{\Gamma(1+x)} $. We have $$ \frac{1}{\Gamma(1+x)}=\sum_{n=0}^\infty ...
9
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1answer
296 views

Complex Factorial Equaling One

For what complex values of $z$ is $$z! =1? $$ Are they even all known? Are there finitely many or infinitely many? (Yes, the trivial $z$ are 0 and 1. )
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3answers
174 views

Scary contour integral, but is also an integral representation for $\Gamma$-function

This problem is supposed to be from an old Acta Mathematica volume I circa 1880's, and is attributed to Bourguet. By using a parabola with its focus on the origin as a contour, show that: ...
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2answers
27 views

Convergence of $\Gamma(z)=\lim_{n\to\infty}\frac{n^zn!}{\prod_{m=0}^n(z+m)}$

This question is a place to store proofs of the convergence of Euler's product formula for the gamma function: $$\Gamma(z)=\lim_{n\to\infty}\frac{n^zn!}{\prod_{m=0}^n(z+m)}$$ which is convergent for ...
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1answer
63 views

Approximation for lower incomplete gamma function

Does any one know approximations for the lower incomplete gamma function $\gamma(a,bx)$. The problem is this: I want to find the quantile function for the CDF of the gamma distribution. The CDF of the ...
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2answers
34 views

Integral related to the gamma function: $\int_0^{\infty} y^{\alpha-1}e^{-y}dy=(\alpha-1)\int_0^{\infty} y^{\alpha-2}e^{-y}dy$

I have difficulty with the gamma function: $\int_0^{\infty} y^{\alpha-1}e^{-y}dy=(\alpha-1)\int_0^{\infty} y^{\alpha-2}e^{-y}dy$ How do we go from left to right? Thanks!
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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 ...
3
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1answer
48 views

Show that $\frac{\Gamma(\frac 1 3)^2}{\Gamma(\frac 1 6)}=\frac{\sqrt {\pi}\sqrt[3] 2}{\sqrt 3}$.

Show that $$\frac{\Gamma(\frac 1 3)^2}{\Gamma(\frac 1 6)}=\frac{\sqrt {\pi}\sqrt[3] 2}{\sqrt 3}$$ Since there's $\sqrt {\pi}$, I suspect I have to related it to $\Gamma(1/2)$. Please give me some ...