On the series $\sum_{n=-\infty}^{\infty} \frac{\cos nx}{\Gamma\left ( a+n+1 \right ) \Gamma \left ( a-n+1 \right )}$

Question

For the values of $a$ for which the following series makes sense, prove that

$$\sum_{n=-\infty}^{\infty} \frac{\cos nx}{\Gamma\left ( a+n+1 \right ) \Gamma \left ( a-n+1 \right )} = \frac{\left ( 2 \cos \frac{x}{2} \right )^{2a}}{\Gamma\left ( 2a+1 \right )}$$

I ran into this series recently and I have no idea how to attack it. How would you go to prove the identity?

Answer 1

Indeed, following Jack's comment we have

\begin{align*} \sum_{n=-\infty}^{\infty} \frac{\cos nx}{\Gamma\left ( a+n+1 \right ) \Gamma \left ( a-n+1 \right )} &= \sum_{n=-\infty}^{\infty} \frac{\cos nx}{\left ( a+n \right )! \left ( a-n \right )!} \\ &=\frac{1}{\left (2a \right )!}\sum_{n=-\infty}^{\infty} \binom{2a}{a+n} \cos nx \\ &=\frac{1}{\left (2a \right )!}\mathfrak{Re} \left ( \sum_{n=-\infty}^{\infty} \binom{2a}{a+n} e^{inx} \right ) \\ &= \frac{\left ( 2 \cos \frac{x}{2} \right )^{2a}}{\left ( 2a \right )!} \\ &= \frac{\left ( 2 \cos \frac{x}{2} \right )^{2a}}{\Gamma\left ( 2a+1 \right )} \end{align*}

from the extended binomial theorem.