Lerche–Newberger sum rule

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 J_α of the first kind. It states that if μ is any non-integer complex number, $\scriptstyle \gamma \in (0,1]$ , and Re(α + β) > −1, then

\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

Newberger, Barry S. (1982), "New sum rule for products of Bessel functions with application to plasma physics", J. Math. Phys., 23 (7): 1278, doi:10.1063/1.525510.
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.
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.
Lerche, I. (1966), "Transverse waves in a relativistic plasma", Physics of Fluids, 9 (6): 1073, doi:10.1063/1.1761804.
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, doi:10.1063/1.2769968.
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.
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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[1] Newberger, Barry S. (1982), "New sum rule for products of Bessel functions with application to plasma physics", J. Math. Phys., 23 (7): 1278, 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, 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, doi:10.1063/1.526282.

[4] Lerche, I. (1966), "Transverse waves in a relativistic plasma", Physics of Fluids, 9 (6): 1073, 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, 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.