Lerche–Newberger sum rule

Template:Short description The Lerche–Newberger, or Newberger, sum rule, discovered by B. S. Newberger in 1982, finds the sum of certain infinite series involving Bessel functions J_α of the first kind. It states that if μ is any non-integer complex number, $\gamma \in (0,1]$, and Re(α + β) > −1, then

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

Newberger's formula generalizes a formula of this type proven by Lerche in 1966; Newberger discovered it independently. Lerche's formula has γ =1; both extend a standard rule for the summation of Bessel functions, and are useful in plasma physics.

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