Infinite Product of Product of Sequence of n plus alpha over Sequence of n plus beta

Theorem

 * $\ds \prod_{n \mathop \ge 1} \dfrac {\paren {n + \alpha_1} \cdots \paren {n + \alpha_k} } {\paren {n + \beta_1} \cdots \paren {n + \beta_k} } = \dfrac {\map \Gamma {1 + \beta_1} \cdots \map \Gamma {1 + \beta_k} } {\map \Gamma {1 + \alpha_1} \cdots \map \Gamma {1 + \alpha_k} }$

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 would have $0$ as its denominator, and so would be undefined.

We have from the Euler form of the Gamma function that:
 * $\map \Gamma {1 + \beta_i} = \ds \lim_{m \mathop \to \infty} \dfrac {m^{1 + \beta_i} m!} {\paren {1 + \beta_i} \paren {2 + \beta_i} \cdots \paren {m + 1 + \beta_i} }$

and so the can be written as: