\[A_{k}^{\ell}(\eta)=\dfrac{(-\mathrm{i})^{k-\ell-1}}{(k-\ell-1)!}\*{{}_{2}F_{1}%
}\left(\ell+1-k,\ell+1-\mathrm{i}\eta;2\ell+2;2\right).\]