Kapteyn series

Let us suppose that the Taylor series of f reads as : f(z)=\sum_{n=0}^\infty a_nz^n. Then the \alpha_n coefficients in the Kapteyn expansion of f can be determined as follows. : \begin{align} \alpha_0 &= a_0,\\ \alpha_n &= \frac14\sum_{k=0}^{\left\lfloor\frac{n}2 \right\rfloor}\frac{(n-2k)^2(n-k-1)!}{k!(n/2)^{(n-2k+1)}}a_{n-2k}\quad(n\ge1). \end{align} ==Examples==

The Kapteyn series of the powers of z are found by Kapteyn himself: : \left(\frac{z}{2}\right)^{n}=n^{2} \sum_{m=0}^\infty \frac{(n+m-1)!}{(n+2 m)^{n+1}\, m!} J_{n+2 m}\{(n+2 m) z\}\quad(z\in D_1). For n = 1 it follows (see also ) : z = 2 \sum_{k=0}^\infty \frac{J_{2k+1}((2k+1)z)}{(2k+1)^2}, and for n = 2 : z^2 = 2 \sum_{k=1}^\infty \frac{J_{2k}(2kz)}{k^2}. Furthermore, inside the region D_1, : \frac{1}{1-z} = 1 + 2 \sum_{k=1}^\infty J_k(kz). == See also ==