Lerche–Newberger sum rule

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, 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.[4][5][6][7]

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. 23, 1278 (1982)]", 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" [Phys. Plasmas 14, 092103 (2007)]", 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'" [Phys. Plasmas 15, 024701 (2008)]", Physics of Plasmas, 15 (2): 024702, doi:10.1063/1.2839770.


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