The Lerche–Newberger, or Newberger, sum rule, discovered by B. S. Newberger in 1982, finds the sum of certain
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
involving
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrar ...
s ''J''
''α'' of the first kind.
It states that if ''μ'' is any non-integer
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
,
, and Re(''α'' + ''β'') > −1, then
:
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), plasma physics.
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References
Special functions
Mathematical identities
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