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A Gamma summation

Let a \notin \mathbb{Z} and a > \frac{1}{2}. Prove that

    \[\sum_{n=0}^{\infty} \frac{2}{\Gamma \left ( a + n \right ) \Gamma \left ( a - n \right )} = \frac{2^{2a-2}}{\Gamma \left ( 2a - 1 \right )} + \frac{1}{\Gamma^2 (a)}\]

Solution

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