\[B_{k}(\zeta)=-\zeta^{-\frac{1}{2}}\sum_{j=0}^{2k+1}(\tfrac{3}{2})^{j}u_{j}%
\zeta^{-3j/2}U_{2k-j+1}\left((1-z^{2})^{-\frac{1}{2}}\right),\]