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Zbl 0559.33003
Vavreck, Andrew N.; Thompson, William jun.
Some novel infinite series of spherical Bessel functions. (English)
[J] Q. Appl. Math. 42, 321-324 (1984). ISSN 0033-569X; ISSN 1552-4485/e

Take two spherical coordinate systems (r,$\theta$,$\phi)$, (r',$\theta$ ',$\phi$ '), with axes parallel and the origin of the second system at $(r\sb 0,0,0)$ with respect to the first. Take an axisymmetric spherical wave function $h\sb n(kr)P\sb n(\cos \theta)$ in the first system; this is expressible as a known summation, over the indices $\nu$ and p, of terms involving $j\sb{\nu}(kr\sb 0)h\sb p(kr')P\sb p(\cos \theta ').$ \par 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 $j\sb n$-functions which sum to 1 or 0. The simplest is the result $\sum\sp{\infty}\sb{0}(2m+1)j\sp 2\sb m(\xi)=1,$ which (as the author remarks) is documented, but most of the others are apparently new and the idea is clearly capable of extension.
[F.M.Arscott]

MSC 2000:: *33C10 Cylinder functions, etc.

Keywords: spherical Bessel functions; spherical wave function

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