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I saw the beta function:

$$ \frac{\Gamma(r)\Gamma(s)}{\Gamma(r+s)}= \int_0^1 t^{(r-1)}(1-t)^{(s-1)} dt $$

and got me wondering if I could do something similar the product of 3 or more gamma functions. What I mean is is there a nice form to express:

$$ \frac{\Gamma(r)\Gamma(s)\Gamma(k)}{\Gamma(r+s+k)} $$

as a nice integral?

Chappers
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drewdles
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1 Answers1

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Yes. In general, let $\Sigma = \{(x_1,\ldots, x_n) \in \Bbb R^n \, |\, x_i\ge 0, \sum x_i \le 1\}$. If $\operatorname{Re}(s_i) > 0$ for all $i$, then

$$\int\cdots \int_\Sigma x_1^{s_1 - 1}x_2^{s_2 - 1}\cdots x_n^{s_n - 1}\, dx_1\cdots\, dx_n = \frac{\Gamma(s_1)\cdots \Gamma(s_n)}{\Gamma(1 + s_1 + \cdots + s_n)}.$$

This can be proven by induction.

Using the relation $\Gamma(z + 1) = z \Gamma(z)$, we represent

$$\frac{\Gamma(s_1) \cdots \Gamma(s_n)}{\Gamma(s_1 + \cdots + s_n)} = s\int \cdots \int_{\Sigma} x_1^{s_1 - 1}\cdots x_n^{s_n - 1} dx_1\ldots dx_n$$

where $s = s_1 + \cdots + s_n$.

kobe
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