Lerche–Newberger sum rule

The Lerche–Newberger, or Newberger, sum rule, discovered by B. S. Newberger in 1982, finds the sum of certain infinite series involving Bessel functions Jα of the first kind. It states that if μ is any non-integer complex number, γ ∈ ( 0 , 1 ] {\displaystyle \scriptstyle \gamma \in (0,1]} , and Re(α + β) > −1, then ∑ n = − ∞ ∞ ( − 1 ) n J α − γ n ( z ) J β + γ n ( z ) n + μ = π sin ⁡ μ π J α + γ μ ( z ) J β − γ μ ( z ) .

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