Newberger's summation formula

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Newberger's summation forumula is a sum rule discovered 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)

where J is the Bessel function of the first kind and γ is the Euler-Mascheroni constant.


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