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Some novel infinite series of spherical Bessel functions. (English) Zbl 0559.33003

Take two spherical coordinate systems (r,θ,φ), (r’,θ ’,φ ’), with axes parallel and the origin of the second system at (r 0 ,0,0) with respect to the first. Take an axisymmetric spherical wave function h n (kr)P n (cosθ) in the first system; this is expressible as a known summation, over the indices ν and p, of terms involving j ν (kr 0 )h p (kr ' )P p (cosθ ' )·

By transforming from the first system to the second and then back again, the author obtains an identity, from which may be extracted a number of series involving j n -functions which sum to 1 or 0. The simplest is the result 0 (2m+1)j m 2 (ξ)=1, which (as the author remarks) is documented, but most of the others are apparently new and the idea is clearly capable of extension.

Reviewer: F.M.Arscott
MSC:
33C10Bessel and Airy functions, cylinder functions, 0 F 1