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On some infinite series involving the zeros of Bessel functions of the first kind

Published online by Cambridge University Press:  18 May 2009

Ian N. Sneddon
Ian N. Sneddon
Affiliation:
The University Glasgow
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In this paper we shall be concerned with the derivation of simple expressions for the sums of some infinite series involving the zeros of Bessel functions of the first kind. For instance, if we denote by γv, n (n = l, 2, 3,…) the positive zeros of Jv(z), then, in certain physical applications, we are interested in finding the values of the sums

and

where m is a positive integer. In § 4 of this paper we shall derive a simple recurrence relation for S2m,v which enables the value of any sum to be calculated as a rational function of the order vof the Bessel function. Similar results are given in § 5 for the sum T2m,v.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 4 , Issue 3 , January 1960 , pp. 144 - 156
Copyright
Copyright © Glasgow Mathematical Journal Trust 1960

References

1.Forsyth, A. R., The expression of Bessel functions of positive order as products, and their inverse powers as sums of rational fractions, Messenger of Math., 50, (1920–1921), 129–149.Google Scholar
2.Nielson, N., Die Zylinderfunktionen und ihre Anwendungen (Leipzig, 1904).Google Scholar
3.Buchholz, H., Bemerkungen zu einer Entwicklungsformel aus der Theorie der Zylinderfunktionen, Z. Angew. Math. u. Mech. 25–27 (1947), 245–252.Google Scholar
4.Erdélyi, A. et al. , Higher transcendental functions (New York, 1953), Vol. II, 60–61.Google Scholar
5.Watson, G. N., A treatise on the theory of Bessel functions, 2nd edn, (Cambridge, 1944).Google Scholar