A hypergeometric series

Let \alpha, \beta \in \mathbb{R} such that 0<\alpha<\beta. Evaluate the series:

    \[\mathcal{S} = \sum_{n=0}^{\infty} \frac{\alpha \left ( \alpha+1 \right ) \cdots \left ( \alpha+n \right )}{\left (\beta+1 \right )\left ( \beta+2 \right )\cdots\left ( \beta+n \right )}\]

Solution

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