Kapteyn Series

A series of the form

$$\sum_{n=0}^\infty \alpha_nJ_{\nu+n}[(\nu+n)z],$$

where $J_n(z)$ is a Bessel Function of the First Kind. Examples include Kapteyn's original series

$${1\over 1-z}=1+2\sum_{n=1}^\infty J_n(nz)$$

and

$${z^2\over 2(1-z^2)}=\sum_{n=1}^\infty J_{2n}(2nz).$$

See also Bessel Function of the First Kind, Neumann Series (Bessel Function)

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1473, 1980.