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2011-11-07T05:59:25Z

74.14.209.124: /* See also */

The '''Lerche–Newberger''', or '''Newberger''', '''sum rule''', discovered by B. S. Newberger in 1982,<ref>
{{citation
| last = Newberger
| first = Barry S.
| title = New sum rule for products of Bessel functions with application to plasma physics
| journal = J. Math. Phys.
| volume = 23
| pages = 1278
| year = 1982
| doi = 10.1063/1.525510
| issue = 7}}.</ref><ref>
{{citation
| last = Newberger
| first = Barry S.
| title = Erratum: New sum rule for products of Bessel functions with application to plasma physics [J. Math. Phys. '''23''', 1278 (1982)]
| journal = J. Math. Phys.
| volume = 24
| pages = 2250
| year = 1983
| doi = 10.1063/1.525940
| issue = 8}}.
</ref><ref>
{{citation
| first1 = M. | last1 = Bakker | first2 = N. M. | last2 = Temme
| title = Sum rule for products of Bessel functions: Comments on a paper by Newberger
| journal = J. Math. Phys. | volume = 25 | page = 1266 | year = 1984 | doi = 10.1063/1.526282
| issue = 5}}.
</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]].
It states that if ''μ'' is any non-integer [[complex number]], <math>\scriptstyle\gamma \in (0,1]</math>, and '''Re'''(''α'' + ''β'') > −1, then

:<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>

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>
{{citation
| last = Lerche | first = I.
| title = Transverse waves in a relativistic plasma
| journal = Physics of Fluids
| volume = 9 | page = 1073 | year = 1966 | doi = 10.1063/1.1761804
| issue = 6}}.
</ref><ref>
{{citation
| first1 = Hong | last1 = Qin | first2 = Cynthia K. | last2 = Phillips | first3 = Ronald C. | last3 = Davidson
| title = A new derivation of the plasma susceptibility tensor for a hot magnetized plasma without infinite sums of products of Bessel functions
| journal = Physics of Plasmas | volume = 14 | page = 092103 | year = 2007 | doi = 10.1063/1.2769968
| issue = 9}}.
</ref><ref>
{{citation
| first1 = I. | last1 = Lerche | first2 = R. | last2 = Schlickeiser | first3 = R. C. | last3 = Tautz
| 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)]
| journal = Physics of Plasmas | volume = 15 | page = 024701 | year = 2008 | doi = 10.1063/1.2839769
| issue = 2}}.
</ref><ref>
{{citation
| first1 = Hong | last1 = Qin | first2 = Cynthia K. | last2 = Phillips | first3 = Ronald C. | last3 = Davidson
| 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)]
| journal = Physics of Plasmas | volume = 15 | page = 024702 | year = 2008 | doi = 10.1063/1.2839770
| issue = 2}}.
</ref>

==References==
{{reflist|2}}

{{DEFAULTSORT:Lerche-Newberger sum rule}}
[[Category:Special functions]]
[[Category:Mathematical identities]]

{{mathanalysis-stub}}

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