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Lerche–Newberger sum rule

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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, 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, γ ∈ ( 0 , 1 ] {\displaystyle \scriptstyle \gamma \in (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 ) . {\displaystyle \sum _{n=-\infty }^{\infty }{\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).}

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

  1. Newberger, Barry S. (1982), "New sum rule for products of Bessel functions with application to plasma physics", J. Math. Phys., 23 (7): 1278–1281, Bibcode:1982JMP....23.1278N, doi:10.1063/1.525510.
  2. Newberger, Barry S. (1983), "Erratum: New sum rule for products of Bessel functions with application to plasma physics ", J. Math. Phys., 24 (8): 2250, Bibcode:1983JMP....24.2250N, doi:10.1063/1.525940.
  3. 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, Bibcode:1984JMP....25.1266B, doi:10.1063/1.526282.
  4. Lerche, I. (1966), "Transverse waves in a relativistic plasma", Physics of Fluids, 9 (6): 1073, Bibcode:1966PhFl....9.1073L, doi:10.1063/1.1761804.
  5. Qin, Hong; Phillips, Cynthia K.; Davidson, Ronald C. (2007), "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, Bibcode:2007PhPl...14i2103Q, doi:10.1063/1.2769968.
  6. 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" ", Physics of Plasmas, 15 (2): 024701, doi:10.1063/1.2839769.
  7. Qin, Hong; Phillips, Cynthia K.; Davidson, Ronald C. (2008), "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'" ", Physics of Plasmas, 15 (2): 024702, doi:10.1063/1.2839770.


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