Infinite Product of Product of Sequence of n plus alpha over Sequence of n plus beta
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Theorem
- $\displaystyle \prod_{n \mathop \ge 1} \dfrac {\left({n + \alpha_1}\right) \cdots \left({n + \alpha_k}\right)} {\left({n + \beta_1}\right) \cdots \left({n + \beta_k}\right)} = \dfrac {\Gamma \left({1 + \beta_1}\right) \cdots\Gamma \left({1 + \beta_1}\right)} {\Gamma \left({1 + \alpha_1}\right) \cdots\Gamma \left({1 + \alpha_k}\right)}$
where:
- $\alpha_1 + \cdots + \alpha_k = \beta_1 + \cdots + \beta_k$
- none of the $\beta$s is a negative integer.
Proof
First we note that if any of the $\beta$s is a negative integer, the left hand side would have $0$ as its denominator, and so would be undefined.
We have from the Euler form of the Gamma function that:
- $\Gamma \left({1 + \beta_i}\right) = \displaystyle \lim_{m \mathop \to \infty} \dfrac {m^{1 + \beta_i} m!} {\left({1 + \beta_i}\right) \left({2 + \beta_i}\right) \cdots \left({m + 1 + \beta_i}\right)}$
and so the right hand side can be written as:
\(\displaystyle \) | \(\) | \(\displaystyle \dfrac {\left({\displaystyle \lim_{m \mathop \to \infty} \dfrac {m^{1 + \beta_1} m^{1 + \beta_2} \cdots m^{1 + \beta_k} \left({m!}\right)^k} {\displaystyle \prod_{1 \mathop \le n \mathop \le m + 1} \left({n + \beta_1}\right) \left({n + \beta_2}\right) \cdots \left({n + \beta_k}\right)} }\right)} {\left({\displaystyle \lim_{m \mathop \to \infty} \dfrac {m^{1 + \alpha_1} m^{1 + \alpha_2} \cdots m^{1 + \alpha_k} \left({m!}\right)^k} {\displaystyle \prod_{1 \mathop \le n \mathop \le m + 1} \left({n + \alpha_1}\right) \left({n + \alpha_2}\right) \cdots \left({n + \alpha_k}\right)} }\right)}\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \lim_{m \mathop \to \infty} \dfrac {\displaystyle \prod_{1 \mathop \le n \mathop \le m + 1} \left({n + \alpha_1}\right) \left({n + \alpha_2}\right) \cdots \left({n + \alpha_k}\right) m^k m^{\beta_1 + \beta_2 + \cdots + \beta_k} \left({m!}\right)^k} {\displaystyle \prod_{1 \mathop \le n \mathop \le m + 1} \left({n + \beta_1}\right) \left({n + \beta_2}\right) \cdots \left({n + \beta_k}\right) m^k m^{\alpha_1 + \alpha_2 + \cdots + \alpha_k} \left({m!}\right)^k}\) | |||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \lim_{m \mathop \to \infty} \dfrac {\displaystyle \prod_{1 \mathop \le n \mathop \le m + 1} \left({n + \alpha_1}\right) \left({n + \alpha_2}\right) \cdots \left({n + \alpha_k}\right) m^{\beta_1 + \beta_2 + \cdots + \beta_k} } {\displaystyle \prod_{1 \mathop \le n \mathop \le m + 1} \left({n + \beta_1}\right) \left({n + \beta_2}\right) \cdots \left({n + \beta_k}\right) m^{\alpha_1 + \alpha_2 + \cdots + \alpha_k} }\) | simplifying slightly | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \lim_{m \mathop \to \infty} \displaystyle \prod_{1 \mathop \le n \mathop \le m + 1} \dfrac {\left({n + \alpha_1}\right) \left({n + \alpha_2}\right) \cdots \left({n + \alpha_k}\right)} {\left({n + \beta_1}\right) \left({n + \beta_2}\right) \cdots \left({n + \beta_k}\right)}\) | as $\alpha_1 + \cdots + \alpha_k = \beta_1 + \cdots + \beta_k$ | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \displaystyle \prod_{n \mathop \ge 1} \dfrac {\left({n + \alpha_1}\right) \left({n + \alpha_2}\right) \cdots \left({n + \alpha_k}\right)} {\left({n + \beta_1}\right) \left({n + \beta_2}\right) \cdots \left({n + \beta_k}\right)}\) |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: Exercise $17$