# Tagged Questions

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

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### Intuition for the definition of the Gamma function?

In these notes by Terence Tao is a proof of Stirling's formula. I really like most of it, but at a crucial step he uses the integral identity $$n! = \int_{0}^{\infty} t^n e^{-t} dt$$ coming from ...
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### Why is Euler's Gamma function the “best” extension of the factorial function to the reals?

There are lots (an infinitude) of smooth functions that coincide with f(n)=n! on the integers. Is there a simple reason why Euler's Gamma function $\Gamma (z) = \int_0^\infty t^{z-1} e^t dt$ is "best"...
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### Integral $\int_{-\infty}^\infty\frac{\Gamma(x)\,\sin(\pi x)}{\Gamma\left(x+a\right)}\,dx$

I would like to evaluate this integral: $$\mathcal F(a)=\int_{-\infty}^\infty\frac{\Gamma(x)\,\sin(\pi x)}{\Gamma\left(x+a\right)}\,dx,\quad a>0.\tag1$$ For all $a>0$ the integrand is a smooth ...
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### Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n}$

Is there a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$ This is a ...
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### Continuous generalization of $\sum_{k=0}^n {n \choose k} = 2^n$?

We know that $$\sum_{k=0}^n {n \choose k} = 2^n.$$ A continuous generalization of the formula would be $$\int_0^{n+1} \frac{\Gamma(n+1)}{\Gamma(n-x+1) \Gamma(x+1)} dx = 2^n?,$$ but this is incorrect ...
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### On the zeta sum $\sum_{n=1}^\infty[\zeta(5n)-1]$ and others

For p = 2, we have, \begin{align}&\sum_{n=1}^\infty[\zeta(pn)-1] = \frac{3}{4}\end{align} It seems there is a general form for odd p. For example, for p = 5, define $z_5 = e^{\pi i/5}$. Then, ...
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### Integral that arises from the derivation of Kummer's Fourier expansion of $\ln{\Gamma(x)}$

I am trying to prove that for $0<x<1$, $$\color{blue}{\ln{\Gamma(x)}=\frac{1}{2}\ln(2\pi)+\sum^\infty_{n=1}\left\{\frac{1}{2n}\cos(2\pi nx)+\frac{\gamma+\ln(2\pi n)}{n\pi}\sin(2\pi nx)\right\}}$$...
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### Quotient of gamma functions?

I'm sorry if this is a simple question, but this page on Wolfram Research states that it follows from Stirling's formula that: $$\frac{\Gamma(x+\beta)}{\Gamma(x)} \approx x^\beta$$ for large $x$, ...
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### Gamma $\Gamma$ meets $\gamma$

I am looking for a proofs of the following limits: $$\lim_{x \to \infty} \Gamma \left(1+\frac{1}{x} \right)^x = e^{-\gamma}.$$ I find this limit interesting as it relates the gamma function $\Gamma$...