Lerche–Newberger sum rule
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Short description: Finds the sum of certain infinite series involving Bessel functions of the first kind
The Lerche–Newberger, or Newberger, sum rule, discovered by B. S. Newberger in 1982,[1][2][3] 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, [math]\displaystyle{ \scriptstyle\gamma \in (0,1] }[/math], and Re(α + β) > −1, then
- [math]\displaystyle{ \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.[4][5][6][7]
References
- ↑ Newberger, Barry S. (1982), "New sum rule for products of Bessel functions with application to plasma physics", J. Math. Phys. 23 (7): 1278–1281, doi:10.1063/1.525510, Bibcode: 1982JMP....23.1278N.
- ↑ Newberger, Barry S. (1983), "Erratum: New sum rule for products of Bessel functions with application to plasma physics [J. Math. Phys. 23, 1278 (1982)]", J. Math. Phys. 24 (8): 2250, doi:10.1063/1.525940, Bibcode: 1983JMP....24.2250N.
- ↑ Bakker, M.; Temme, N. M. (1984), "Sum rule for products of Bessel functions: Comments on a paper by Newberger", J. Math. Phys. 25 (5): 1266, doi:10.1063/1.526282, Bibcode: 1984JMP....25.1266B.
- ↑ Lerche, I. (1966), "Transverse waves in a relativistic plasma", Physics of Fluids 9 (6): 1073, doi:10.1063/1.1761804, Bibcode: 1966PhFl....9.1073L.
- ↑ "A new derivation of the plasma susceptibility tensor for a hot magnetized plasma without infinite sums of products of Bessel functions", Physics of Plasmas 14 (9): 092103, 2007, doi:10.1063/1.2769968, Bibcode: 2007PhPl...14i2103Q.
- ↑ Lerche, I.; Schlickeiser, R.; Tautz, R. C. (2008), "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)]", Physics of Plasmas 15 (2): 024701, doi:10.1063/1.2839769.
- ↑ "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)]", Physics of Plasmas 15 (2): 024702, 2008, doi:10.1063/1.2839770.
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