# §33.6 Power-Series Expansions in $\rho$

 33.6.1 $\mathop{F_{\ell}\/}\nolimits\!\left(\eta,\rho\right)=\mathop{C_{\ell}\/}% \nolimits\!\left(\eta\right)\sum_{k=\ell+1}^{\infty}A_{k}^{\ell}(\eta)\rho^{k},$
 33.6.2 ${\mathop{F_{\ell}\/}\nolimits^{\prime}}\!\left(\eta,\rho\right)=\mathop{C_{% \ell}\/}\nolimits\!\left(\eta\right)\sum_{k=\ell+1}^{\infty}kA_{k}^{\ell}(\eta% )\rho^{k-1},$

where $A_{\ell+1}^{\ell}=1$, $A_{\ell+2}^{\ell}=\eta/(\ell+1)$, and

 33.6.3 $(k+\ell)(k-\ell-1)A_{k}^{\ell}=2\eta A_{k-1}^{\ell}-A_{k-2}^{\ell},$ $k=\ell+3,\ell+4,\dots$,

or in terms of the hypergeometric function (§§15.1, 15.2(i)),

 33.6.4 $A_{k}^{\ell}(\eta)=\dfrac{(-i)^{k-\ell-1}}{(k-\ell-1)!}\*\mathop{{{}_{2}F_{1}}% \/}\nolimits\!\left(\ell+1-k,\ell+1-i\eta;2\ell+2;2\right).$
 33.6.5 $\mathop{{H^{\pm}_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)=\frac{e^{\pm i% \mathop{{\theta_{\ell}}\/}\nolimits\!\left(\eta,\rho\right)}}{(2\ell+1)!% \mathop{\Gamma\/}\nolimits\!\left(-\ell+i\eta\right)}\left(\sum_{k=0}^{\infty}% \frac{\left(a\right)_{k}}{\left(2\ell+2\right)_{k}k!}(\mp 2i\rho)^{a+k}\left(% \mathop{\ln\/}\nolimits\!\left(\mp 2i\rho\right)+\mathop{\psi\/}\nolimits\!% \left(a+k\right)-\mathop{\psi\/}\nolimits\!\left(1+k\right)-\mathop{\psi\/}% \nolimits\!\left(2\ell+2+k\right)\right)-\sum_{k=1}^{2\ell+1}\frac{(2\ell+1)!(% k-1)!}{(2\ell+1-k)!\left(1-a\right)_{k}}(\mp 2i\rho)^{a-k}\right),$

where $a=1+\ell\pm i\eta$ and $\mathop{\psi\/}\nolimits\!\left(x\right)={\mathop{\Gamma\/}\nolimits^{\prime}}% \!\left(x\right)/\mathop{\Gamma\/}\nolimits\!\left(x\right)$5.2(i)).

The series (33.6.1), (33.6.2), and (33.6.5) converge for all finite values of $\rho$. Corresponding expansions for ${\mathop{{H^{\pm}_{\ell}}\/}\nolimits^{\prime}}\!\left(\eta,\rho\right)$ can be obtained by combining (33.6.5) with (33.4.3) or (33.4.4).