Lerche–Newberger sum rule

In mathematics, the Lerche–Newberger sum rule, or Newberger sum rule, discovered by B. S. Newberger in 1982,^[1][2][3] finds the sum of certain infinite series involving Bessel functions ${\displaystyle J_{\alpha }}$ of the first kind. It states that if ${\displaystyle \mu }$ is any non-integer complex number, ${\displaystyle \gamma \in (0,1]}$, and ${\displaystyle \Re (\alpha +\beta )>-1}$, then

${\displaystyle \sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}J_{\alpha -\gamma n}(z)J_{\beta +\gamma n}(z)}{n+\mu }}={\frac {\pi }{\sin \mu \pi }}J_{\alpha +\gamma \mu }(z)J_{\beta -\gamma \mu }(z).}$

Newberger's formula generalizes a formula of this type proven by Lerche in 1966. Lerche's formula has ${\displaystyle \gamma =1}$; both extend a standard rule for the summation of Bessel functions, and are useful in plasma physics.^[4][5][6][7]