PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 137, Number 7, July 2009, Pages 2363–2368 S 0002-9939(09)09796-2 Article electronically published on January 22, 2009 INTEGRAL REPRESENTATION FOR NEUMANN SERIES OF BESSEL FUNCTIONS TIBOR K. POG ´ ANY AND ENDRE S ¨ ULI (Communicated by Peter A. Clarkson) Abstract. A closed integral expression is derived for Neumann series of Bessel functions — a series of Bessel functions of increasing order — over the set of real numbers. 1. Introduction and motivation The series (1) N ν (z) := ∞  n=1 α n J ν+n (z), z ∈ C, where ν, α n are constants and J μ signifies the Bessel function of the first kind of order μ, is called a Neumann series [21, Chapter XVI]. Such series owe their name to the fact that they were first systematically considered (for integer μ) by Carl Gottfried Neumann in his important book [15] in 1867; subsequently, in 1877, Leopold Bernhard Gegenbauer extended such series to μ ∈ R (see [21, p. 522]). Neumann series of Bessel functions arise in a number of application areas. For example, in connection with random noise, Rice [18, Eqs. (3.10–3.17)] applied Bennett’s result, (2) ∞  n=1  v a  n J n (ai v)=e v 2 /2  v 0 xe −x 2 /2 J 0 (ai x)dx. Luke [8, pp. 271–288] proved that 1 −  v 0 e −(u+x) J 0 ( 2i √ ux ) dx = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ e −(u+v) ∞  n=0  u v  n/2 J n ( 2i √ uv ) , u < v, 1 − e −(u+v) ∞  n=1  v u  n/2 J n ( 2i √ uv ) , u>v; cf. also [16, Eq. (2a)]. In both of these applications N 0 plays a key role. The function N 0 also appears as a relevant technical tool in the solution of the infinite dielectric wedge problem by Kontorovich–Lebedev transforms [20, §§4, 5]. It also Received by the editors May 31, 2007, and, in revised form, September 22, 2008. 2000 Mathematics Subject Classification. Primary 33C10, 33C20; Secondary 40A05, 44A20. Key words and phrases. Bessel function of the first kind J ν (x), integral representation of series, Neumann series of Bessel functions. The first author was supported in part by Research Project No. 112-2352818-2814 of the Ministry of Sciences, Education and Sports of Croatia. c 2009 American Mathematical Society 2363 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use