# §33.16 Connection Formulas

## §33.16(i) $\mathop{F_{\ell}\/}\nolimits$ and $\mathop{G_{\ell}\/}\nolimits$ in Terms of $\mathop{f\/}\nolimits$ and $\mathop{h\/}\nolimits$

 33.16.1 $\mathop{F_{\ell}\/}\nolimits\!\left(\eta,\rho\right)=\dfrac{(2\ell+1)!\mathop{% C_{\ell}\/}\nolimits\!\left(\eta\right)}{(-2\eta)^{\ell+1}}\mathop{f\/}% \nolimits\!\left(1/\eta^{2},\ell;-\eta\rho\right),$
 33.16.2 $\mathop{G_{\ell}\/}\nolimits\!\left(\eta,\rho\right)=\dfrac{\pi(-2\eta)^{\ell}% }{(2\ell+1)!\mathop{C_{\ell}\/}\nolimits\!\left(\eta\right)}\mathop{h\/}% \nolimits\!\left(1/\eta^{2},\ell;-\eta\rho\right),$

where $\mathop{C_{\ell}\/}\nolimits\!\left(\eta\right)$ is given by (33.2.5) or (33.2.6).

## §33.16(ii) $\mathop{f\/}\nolimits$ and $\mathop{h\/}\nolimits$ in Terms of $\mathop{F_{\ell}\/}\nolimits$ and $\mathop{G_{\ell}\/}\nolimits$ when $\epsilon>0$

When $\epsilon>0$ denote

 33.16.3 $\tau=\epsilon^{1/2}(>0),$ Defines: $\tau$: parameter (locally) Symbols: $\epsilon$: real parameter Referenced by: §33.16(iv) Permalink: http://dlmf.nist.gov/33.16.E3 Encodings: TeX, pMML, png See also: info for 33.16(ii)

and again define $A(\epsilon,\ell)$ by (33.14.11) or (33.14.12). Then for $r>0$

 33.16.4 $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)=\left(\frac{2}{\pi\tau}% \frac{1-e^{-2\pi/\tau}}{A(\epsilon,\ell)}\right)^{\ifrac{1}{2}}\mathop{F_{\ell% }\/}\nolimits\!\left(-1/\tau,\tau r\right),$
 33.16.5 $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)=\left(\frac{2}{\pi\tau}% \frac{A(\epsilon,\ell)}{1-e^{-2\pi/\tau}}\right)^{\ifrac{1}{2}}\mathop{G_{\ell% }\/}\nolimits\!\left(-1/\tau,\tau r\right).$

Alternatively, for $r<0$

 33.16.6 $\displaystyle\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ $\displaystyle=(-1)^{\ell+1}\left(\frac{2}{\pi\tau}\frac{e^{2\pi/\tau}-1}{A(% \epsilon,\ell)}\right)^{\ifrac{1}{2}}\mathop{F_{\ell}\/}\nolimits\!\left(1/% \tau,-\tau r\right),$ 33.16.7 $\displaystyle\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ $\displaystyle=(-1)^{\ell}\left(\frac{2}{\pi\tau}\frac{A(\epsilon,\ell)}{e^{2% \pi/\tau}-1}\right)^{\ifrac{1}{2}}\mathop{G_{\ell}\/}\nolimits\!\left(1/\tau,-% \tau r\right).$

## §33.16(iii) $\mathop{f\/}\nolimits$ and $\mathop{h\/}\nolimits$ in Terms of $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)$ when $\epsilon<0$

When $\epsilon<0$ denote

 33.16.8 $\nu=1/(-\epsilon)^{1/2}(>0),$ Defines: $\nu$: parameter (locally) Symbols: $\epsilon$: real parameter Referenced by: §33.16(v) Permalink: http://dlmf.nist.gov/33.16.E8 Encodings: TeX, pMML, png See also: info for 33.16(iii)
 33.16.9 $\displaystyle\zeta_{\ell}(\nu,r)$ $\displaystyle=\mathop{W_{\nu,\ell+\frac{1}{2}}\/}\nolimits\!\left(2r/\nu\right),$ $\displaystyle\xi_{\ell}(\nu,r)$ $\displaystyle=\realpart{\left(e^{i\pi\nu}\mathop{W_{-\nu,\ell+\frac{1}{2}}\/}% \nolimits\!\left(e^{i\pi}2r/\nu\right)\right)},$ Defines: $\zeta_{\ell}(\nu,r)$: function (locally) and $\xi_{\ell}(\nu,r)$: function (locally) Symbols: $\mathop{W_{\NVar{\kappa},\NVar{\mu}}\/}\nolimits\!\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function, $e$: base of exponential function, $\realpart{}$: real part, $\ell$: nonnegative integer, $r$: real variable and $\nu$: parameter Referenced by: §33.16(v), §33.21(i) Permalink: http://dlmf.nist.gov/33.16.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: info for 33.16(iii)

and again define $A(\epsilon,\ell)$ by (33.14.11) or (33.14.12). Then for $r>0$

 33.16.10 $\displaystyle\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ $\displaystyle=(-1)^{\ell}\nu^{\ell+1}\left(-\frac{\mathop{\cos\/}\nolimits\!% \left(\pi\nu\right)\zeta_{\ell}(\nu,r)}{\mathop{\Gamma\/}\nolimits\!\left(\ell% +1+\nu\right)}+\frac{\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)\mathop{% \Gamma\/}\nolimits\!\left(\nu-\ell\right)\xi_{\ell}(\nu,r)}{\pi}\right),$ 33.16.11 $\displaystyle\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ $\displaystyle=(-1)^{\ell}\nu^{\ell+1}A(\epsilon,\ell)\left(\frac{\mathop{\sin% \/}\nolimits\!\left(\pi\nu\right)\zeta_{\ell}(\nu,r)}{\mathop{\Gamma\/}% \nolimits\!\left(\ell+1+\nu\right)}+\frac{\mathop{\cos\/}\nolimits\!\left(\pi% \nu\right)\mathop{\Gamma\/}\nolimits\!\left(\nu-\ell\right)\xi_{\ell}(\nu,r)}{% \pi}\right).$

Alternatively, for $r<0$

 33.16.12 $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)=\frac{(-1)^{\ell}\nu^{\ell% +1}}{\pi}\left(-\frac{\pi\xi_{\ell}(-\nu,r)}{\mathop{\Gamma\/}\nolimits\!\left% (\ell+1+\nu\right)}+\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)\mathop{\cos% \/}\nolimits\!\left(\pi\nu\right)\mathop{\Gamma\/}\nolimits\!\left(\nu-\ell% \right)\zeta_{\ell}(-\nu,r)\right),$
 33.16.13 $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)=(-1)^{\ell}\nu^{\ell+1}A(% \epsilon,\ell)\mathop{\Gamma\/}\nolimits\!\left(\nu-\ell\right)\zeta_{\ell}(-% \nu,r)/\pi.$

## §33.16(iv) $\mathop{s\/}\nolimits$ and $\mathop{c\/}\nolimits$ in Terms of $\mathop{F_{\ell}\/}\nolimits$ and $\mathop{G_{\ell}\/}\nolimits$ when $\epsilon>0$

When $\epsilon>0$, again denote $\tau$ by (33.16.3). Then for $r>0$

 33.16.14 $\displaystyle\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ $\displaystyle=(\pi\tau)^{-1/2}\mathop{F_{\ell}\/}\nolimits\!\left(-1/\tau,\tau r% \right),$ $\displaystyle\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)$ $\displaystyle=(\pi\tau)^{-1/2}\mathop{G_{\ell}\/}\nolimits\!\left(-1/\tau,\tau r% \right).$

Alternatively, for $r<0$

 33.16.15 $\displaystyle\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ $\displaystyle=(\pi\tau)^{-1/2}\mathop{F_{\ell}\/}\nolimits\!\left(1/\tau,-\tau r% \right),$ $\displaystyle\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)$ $\displaystyle=(\pi\tau)^{-1/2}\mathop{G_{\ell}\/}\nolimits\!\left(1/\tau,-\tau r% \right).$

## §33.16(v) $\mathop{s\/}\nolimits$ and $\mathop{c\/}\nolimits$ in Terms of $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(z\right)$ when $\epsilon<0$

When $\epsilon<0$ denote $\nu$, $\zeta_{\ell}(\nu,r)$, and $\xi_{\ell}(\nu,r)$ by (33.16.8) and (33.16.9). Also denote

 33.16.16 $K(\nu,\ell)=\left(\nu^{2}\mathop{\Gamma\/}\nolimits\!\left(\nu+\ell+1\right)% \mathop{\Gamma\/}\nolimits\!\left(\nu-\ell\right)\right)^{-1/2}.$ Defines: $K(\nu,\ell)$: factor (locally) Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\ell$: nonnegative integer and $\nu$: parameter Referenced by: §33.16(v) Permalink: http://dlmf.nist.gov/33.16.E16 Encodings: TeX, pMML, png See also: info for 33.16(v)

Then for $r>0$

 33.16.17 $\displaystyle\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ $\displaystyle=\frac{(-1)^{\ell}}{2\nu^{1/2}}\left(\frac{\mathop{\sin\/}% \nolimits\!\left(\pi\nu\right)}{\pi K(\nu,\ell)}\xi_{\ell}(\nu,r)-\mathop{\cos% \/}\nolimits\!\left(\pi\nu\right)\nu^{2}K(\nu,\ell)\zeta_{\ell}(\nu,r)\right),$ $\displaystyle\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)$ $\displaystyle=\frac{(-1)^{\ell}}{2\nu^{1/2}}\left(\frac{\mathop{\cos\/}% \nolimits\!\left(\pi\nu\right)}{\pi K(\nu,\ell)}\xi_{\ell}(\nu,r)+\mathop{\sin% \/}\nolimits\!\left(\pi\nu\right)\nu^{2}K(\nu,\ell)\zeta_{\ell}(\nu,r)\right).$

Alternatively, for $r<0$

 33.16.18 $\displaystyle\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$ $\displaystyle=\frac{(-1)^{\ell+1}}{2^{1/2}}\left(\frac{\nu^{3/2}}{K(\nu,\ell)}% \xi_{\ell}(-\nu,r)-\frac{\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)\mathop{% \cos\/}\nolimits\!\left(\pi\nu\right)}{\pi\nu^{1/2}}K(\nu,\ell)\zeta_{\ell}(-% \nu,r)\right),$ $\displaystyle\mathop{c\/}\nolimits\!\left(\epsilon,\ell;r\right)$ $\displaystyle=\frac{(-1)^{\ell}}{\pi(2\nu)^{1/2}}K(\nu,\ell)\zeta_{\ell}(-\nu,% r).$