Lerche–Newberger sum rule - Alchetron, the free social encyclopedia

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, γ ∈ ( 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 ) .

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.

References

Lerche–Newberger sum rule Wikipedia

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