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| The '''Lerche–Newberger''', or '''Newberger''', '''sum rule''', discovered by B. S. Newberger in 1982,<ref>
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| {{citation
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| | last = Newberger
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| | first = Barry S.
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| | title = New sum rule for products of Bessel functions with application to plasma physics
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| | journal = J. Math. Phys.
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| | volume = 23
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| | pages = 1278
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| | year = 1982
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| | doi = 10.1063/1.525510
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| | issue = 7}}.</ref><ref>
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| {{citation
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| | last = Newberger
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| | first = Barry S.
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| | title = Erratum: New sum rule for products of Bessel functions with application to plasma physics [J. Math. Phys. '''23''', 1278 (1982)]
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| | journal = J. Math. Phys.
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| | volume = 24
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| | pages = 2250
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| | year = 1983
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| | doi = 10.1063/1.525940
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| | issue = 8}}.
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| </ref><ref>
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| {{citation
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| | first1 = M. | last1 = Bakker | first2 = N. M. | last2 = Temme
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| | title = Sum rule for products of Bessel functions: Comments on a paper by Newberger
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| | journal = J. Math. Phys. | volume = 25 | page = 1266 | year = 1984 | doi = 10.1063/1.526282
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| | issue = 5}}.
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| </ref> finds the sum of certain [[infinite series]] involving [[Bessel function]]s ''J''<sub>''α''</sub> [[Bessel function#Bessel functions of the first kind|of the first kind]].
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| It states that if ''μ'' is any non-integer [[complex number]], <math>\scriptstyle\gamma \in (0,1]</math>, and '''Re'''(''α'' + ''β'') > −1, then
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| :<math>\sum_{n=- \infin}^\infin\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).</math> | |
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| 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.<ref>
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| {{citation
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| | last = Lerche | first = I.
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| | title = Transverse waves in a relativistic plasma
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| | journal = Physics of Fluids
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| | volume = 9 | page = 1073 | year = 1966 | doi = 10.1063/1.1761804
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| | issue = 6}}.
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| </ref><ref>
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| {{citation
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| | first1 = Hong | last1 = Qin | first2 = Cynthia K. | last2 = Phillips | first3 = Ronald C. | last3 = Davidson
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| | title = A new derivation of the plasma susceptibility tensor for a hot magnetized plasma without infinite sums of products of Bessel functions
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| | journal = Physics of Plasmas | volume = 14 | page = 092103 | year = 2007 | doi = 10.1063/1.2769968
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| | issue = 9}}.
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| </ref><ref>
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| {{citation
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| | first1 = I. | last1 = Lerche | first2 = R. | last2 = Schlickeiser | first3 = R. C. | last3 = Tautz
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| | title = Comment on "A new derivation of the plasma susceptibility tensor for a hot magnetized plasma without infinite sums of products of Bessel functions" [Phys. Plasmas '''14''', 092103 (2007)]
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| | journal = Physics of Plasmas | volume = 15 | page = 024701 | year = 2008 | doi = 10.1063/1.2839769
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| | issue = 2}}.
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| </ref><ref>
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| {{citation
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| | first1 = Hong | last1 = Qin | first2 = Cynthia K. | last2 = Phillips | first3 = Ronald C. | last3 = Davidson
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| | title = Response to "Comment on `A new derivation of the plasma susceptibility tensor for a hot magnetized plasma without infinite sums of products of Bessel functions'" [Phys. Plasmas 15, 024701 (2008)]
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| | journal = Physics of Plasmas | volume = 15 | page = 024702 | year = 2008 | doi = 10.1063/1.2839770
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| | issue = 2}}.
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| </ref>
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| ==References==
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| {{reflist|2}}
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| {{DEFAULTSORT:Lerche-Newberger sum rule}}
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| [[Category:Special functions]]
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| [[Category:Mathematical identities]]
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| {{mathanalysis-stub}}
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