# Questions tagged [gamma-function]

Questions on the gamma function $\Gamma(z)$ of Euler extending the usual factorial $n!$ for arbitrary argument, and related functions. The Gamma function is a specific way to extend the factorial function to other values using integrals.

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### On moment generating function of generalized gamma distribution

I'm reading An Intermediate Course in Probability by Gut. I am confused about a statement made concerning the generalized gamma distribution and its existence of a moment generating function. I quote: ...
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### Estimation of a gamma function-like integral

A random variable $X$ has a pdf: $$f(x) = \frac{1}{k!} \cdot x^k \cdot e^{-x}$$ Prove that $$P(0<X<2\cdot(k+1)) > \frac{k}{k+1}$$ There are no conditions about $k$, so it can be non-integer. ...
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### Can the gamma function be generalized to quaternions and how? [duplicate]

The gamma function is a generalization of the operator !n. The question is: Can the concept of the gamma function be generalized to quaternion analysis and the use of quaternions, and how?
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### Is there an analog for factorials in division, and if so, what are its applications and properties? [closed]

If we consider a factorial to be an operation/function of iterative multiplication, would it be reasonable to think that something similar for division also exists? If we take this function to be f, ...
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### How to define double factorial for non positive integers?

I studied double factorial which known for natural number $$n!!=n(n-2)!! , 1!!=0!!=1$$ So we have for $n\in N$ $$(2n)!!=2^n n! , (2n+1)!!=\frac{(2n+1)!}{2^n n!}$$ but I found on Math-World formula ...
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### How to show that $1-\sqrt{\dfrac{2}{n}}\dfrac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2})}<\dfrac{1}{4n}$

In the calculation of a problem, I need to show that $1-\sqrt{\dfrac{2}{n}}\dfrac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2})}<\dfrac{1}{4n}$ holds for any positive integer $n$. I got the expansion ...
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### how to use Gauss Multiplication Formula for Gamma function?

I studied Gauss Multiplication Formula which known for $n\in Z^+ \wedge nx\notin Z^-\cup\{0\}$ $$\Gamma(nz)=(2\pi)^{(1-n)/2}n^{nz-(1/2)}\prod_{k=0}^{n-1}\Gamma\left(z+\frac{k}{n}\right)$$ but I didn't ...
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### The sum $s(k)=\sum_{n=1}^\infty\frac{\Gamma(\frac{1}{2^n}+1)}{\Gamma(\frac{1}{2^n}-k)}$ gives weird fractions

I stumbled across a series while playing around with a functional equation, it looks like this: $$s(k)=\sum_{n=1}^\infty\frac{\Gamma(\frac{1}{2^n}+1)}{\Gamma(\frac{1}{2^n}-k)}$$ Mathematica gives ...
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### What is a good way to compute $\Gamma(1/3)$ on a standard pocket calculator?

This question is inspired by this one. The earlier question asks how to calculate a certain integral efficiently with a standard pocket calculator. A fine answer by Travis Willse gives a good result ...
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### Numerical Computation of the Gamma Function for large complex numbers

I'm looking for a method to numerically compute the Gamma function $Γ(z)$ for complex numbers of the form $$z= \frac{1}{2} + it,$$ particularly for large values of $t$. Does anyone know of any ...
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### Gamma distribution tricky problem

Suppose that the time, in hours, it takes to repair a pump is a random variable X which has a gamma distribution with parameters α = 2, β = 1/2. what is the probability that a. it takes at most 1 hour ...
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### Transformation from squared integral to double integral

for context, I saw a friend of mine did this, is he correct? $$[T(1/2)]^2 = 4(\int_0^{\infty} e^{-x^2}dx)^2$$ which becomes: $$[T(1/2)]^2 = 4(\int_0^{\infty} \int_0^{\infty} e^{-x^2} e^{-y^2}dx dy)$$ ...
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### Derivative of incomplete gamma function w.r.t. the first argument

Wolfram Research provides the following formulas for the derivative of the incomplete gamma function: https://functions.wolfram.com/GammaBetaErf/Gamma2/20/01/01/0002/ https://functions.wolfram.com/...
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### Is it possible to find the $n$th derivative of Gamma function?

By repeatedly differentiating $\Gamma(x)$, I noticed that $$\frac{d^{n}}{{dx}^{n}}\Gamma(x)=\sum_{k=0}^{n-1}\binom{n-1}{k}\psi^{(n-k-1)}(x)\,\frac{d^{k}}{{dx}^{k}}\Gamma(x),$$ where $\psi^{(a)}(x)$ is ...
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### Asymptotic expansion / behaviour of integral function at large x [closed]

How can I find the asymptotic expansion of the following integral, i.e. its behavior for large $x$? $$\int_0^x \frac{y^k}{\sqrt{1+y^l}}dy$$ I know that the integral can be solved exactly for certain ...
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### Could there be a function $f$ that satisfies $g(s)=f^{(s)}(x)^s|_{x=1/e}=s!$?

I was thinking about the factorial function today and I wondered: Is there a function $f$ that satisfies $g(s)=f^{(s)}(x)^s|_{x=1/e}=s!$ where the notation is the $s$th derivative of $f$ to the $s$th ...
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### Bounding $\Gamma(s)\Gamma(1-s) \sin (\pi s)$ to prove Euler reflection formula

I was reading lecture notes of the course $18.785$ offered by MIT. The author makes an error and write $|\sin(\pi s)|$ as $\frac{1}{2}|e^{is} - e^{i\overline{s}}|$ as can be seen in the picture. Could ...
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### Is there a recursive identity for the derivative of the gamma function $\Gamma'(x)$ [closed]

For the normal gamma we have: $$\Gamma(x) = (x-1)\Gamma(x-1)$$ For the digamma we have: $$\Psi(x) = \Psi(x-1) + \frac{1}{x-1}$$ Is there something similar for $\Gamma'(x) = f(x-1)$?
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### Proof using the Stirling formula for a limit behaviour

I am trying to understand/prove a lemma which is stated in two papers about branching processes (Lemma 2.2 in "Martingales And Large Deviations For Binary Search Trees" by Jabbour-Hattab and ...
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### $\sum_{n=1}^{\infty} \frac{n 4^n}{(2 n-1)^2(4 n+1)(4 n+3)} \frac{\binom{2 n}{n}}{\binom{4 n}{2 n}}=\frac{4(1+\sqrt{2})}{225}$ [closed]

I want to show that $\sum_{n=1}^{\infty} \frac{n 4^n}{(2 n-1)^2(4 n+1)(4 n+3)} \frac{\binom{2 n}{n}}{\binom{4 n}{2 n}}=\frac{4(1+\sqrt{2})}{225}.$ I tried manipulating the terms in the sum, but that ...
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I am having the following equation: $P_1 = \int_0^{\infty} \gamma(m_0, \frac{\phi m_0}{\Omega_0 h})\cdot h^{m_1-1} \cdot e^{-\frac{m_1}{\Omega_1}h}\text{d}h$ --- (1) where $\gamma(\cdot,\cdot)$ is ...