# Tag Info

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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 ...
• 572
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 $k!$ ...
• 33.2k

### 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. ...
• 3,249
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, $$(a \div b) \div c \not= a \div (b \div c)$$ ...
• 34.5k
1 vote

• 268k
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 ...
• 6,563

• 5,840
Accepted

### 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}\\ ...
• 40.3k
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+...