Neumann Series (Bessel Function)

A series of the form

$$\sum_{n=0}^\infty a_nJ_{\nu+n}(z),$$ (1)

where $\nu$ is a Real and $J_{\nu+n}(z)$ is a Bessel Function of the First Kind. Special cases are

$$z^\nu =2^\nu \Gamma({\textstyle{1\over 2}}\nu+1) \sum_{n=0}^... ... {({\textstyle{1\over 2}}z)^{\nu/2+n}\over n!} J_{\nu/2+n}(z),$$ (2)

where $\Gamma(z)$ is the Gamma Function, and

$$\sum_{n=0}^\infty b_nz^{\nu+n}=\sum_{n=0}^\infty a_n\left({{\textstyle{1\over 2}}z}\right)^{(\nu+n)/2}J_{(\nu+n)/2}(z),$$ (3)

where

$$a_n\equiv \sum_{m=0}^{\left\lfloor{n/2}\right\rfloor } {2^{\... ...{1\over 2}}\nu+{\textstyle{1\over 2}}n-m+1)\over m!} b_{n-2m},$$ (4)

and $\left\lfloor{x}\right\rfloor$ is the Floor Function.

See also Kapteyn Series

References

Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.