Newberger's summation formula

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Newberger's summation forumula is a sum rule disvocered by B. S. Newberger in 1982. It is good for $μ$ nonintegral and $\Re (\alpha + \beta) > -1$ :

$\sum_{n=- \infin}^\infin\frac{(-1)^n J_{\alpha - \gamma n}(z)J_{\beta + \gamma n}(z)}{n+\mu}=\frac{\pi}{\sin \mu \pi}J_{\alpha + \gamma \mu}(z)J_{\beta - \gamma \mu}(z)\quad\quad\quad(1)$