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On moment generating function of generalized gamma distribution

I got $C=\alpha/\Gamma(\beta/\alpha)$. I think $\beta>-1$ is needed for the $r$th moment existence, where $(\beta+r)/\alpha$ will show up in place of $\beta/\alpha$. For example, if $\beta=-1$ and ...
Zack Fisher's user avatar
1 vote
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Estimation of a gamma function-like integral

This is probably not the simplest answer. We need to show that $$ \frac{1}{{k!}}\int_{2k + 2}^{ + \infty } {x^k {\rm e}^{ - x} {\rm d}x} < \frac{1}{{k + 1}} $$ for $k>-1$. (I simply define $k!$ ...
Gary's user avatar
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0 votes

Can the gamma function be generalized to quaternions and how?

The quaternions (1,i,k,l) form an abstract representation of the Lie algebra of the orthogonal group. Its lowest dimensional linear representation are the imaginary multiples of the Pauli matrices. ...
Roland F's user avatar
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1 vote

Is there an analog for factorials in division, and if so, what are its applications and properties?

As some of us have indicated in the comments to the OP, repeated division without grouping is ambiguous because division is not associative. That is, $$ (a \div b) \div c \not= a \div (b \div c) $$ ...
Brian Tung's user avatar
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1 vote

Contour integration of an integral

Partial answer By exploiting the symmetry of the integrand and rewriting it using $(8.4.6)$ (the same is obtained by Mathematica's FunctionExpand), we have $$\...
user170231's user avatar
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2 votes

How can I simplify $\prod_\limits{n=1\atop n\ne m}^{a}\frac{nx - 1}{n - m}:\ ? $

Assuming $m>0$ and $a>m$ Denominator $$\prod\limits_{n=1, n\ne m}^a (n-m)=\prod\limits_{n=1}^{m-1} (n-m)\prod\limits_{n=m+1}^{a} (n-m)$$ Using Pochhammer symbols $$\prod\limits_{n=1}^{m-1} (n-m)=...
Claude Leibovici's user avatar
1 vote

How can I simplify $\prod_\limits{n=1\atop n\ne m}^{a}\frac{nx - 1}{n - m}:\ ? $

$$P_m:=\prod\limits_{n=1, n\ne m}^a (n-m)= (m-1)! (a-m)!(-1)^{m-1}$$ Proof:- for fixed $m$ the first $m-1$ numbers in the product are $(1-m )(2-m)\dots 1 = (-1)^{m-1}(m-1)!$ then the rest numbers in ...
pie's user avatar
  • 6,563
2 votes

How can I simplify $\prod_\limits{n=1\atop n\ne m}^{a}\frac{nx - 1}{n - m}:\ ? $

First of all, WolframAlpha was wrong in that it didn't understand the condition $n\ne m$: what's true is $$ \prod_{n=1}^a \biggl(\frac{nx-1}{n-z}\biggr) = \frac{x^{a}\Gamma(1-z)\Gamma(1+a-x^{-1})}{\...
Greg Martin's user avatar
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0 votes
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How can we calculate this integral if this diverges?

In the case $s=5/2$ and $s=7/2$, the integral is divergent. The formula makes sense when the integral is convergent. For example, when $s=5/2$ you have: $$\frac{x^{3/2}}{1+x}\sim x^{1/2} \quad \text{...
Sine of the Time's user avatar
6 votes
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How to prove the following integral inequality problem

The desired inequality is written as $$\int_0^1 (1 - x^a)^b b^{1/a}\,\mathrm{d} x + \frac{1}{1 + a} \ge 1.$$ We have \begin{align*} &\int_0^1 (1 - x^a)^b b^{1/a}\,\mathrm{d} x + \frac{1}{1 + a}\\ ...
River Li's user avatar
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4 votes

How to prove the following integral inequality problem

A generalization of the Gautschi's inequality states that (see https://math.stackexchange.com/a/2089967/326159) $$\frac{1}{sn^{s-1}} \leq \frac{\Gamma(n+1)}{s\Gamma(n+s)} = \frac{\Gamma(n+1)}{\Gamma(n+...
user326159's user avatar
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