# Β§33.16 Connection Formulas

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

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

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

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

When $\epsilon>0$ denote

 33.16.3 $\tau=\epsilon^{1/2}(>0),$ β Defines: $\tau$: parameter (locally) Symbols: $\epsilon$: real parameter Referenced by: Β§33.16(iv) Permalink: http://dlmf.nist.gov/33.16.E3 Encodings: TeX, pMML, png See also: Annotations for Β§33.16(ii), Β§33.16 and Ch.33

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

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

Alternatively, for $r<0$

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

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

When $\epsilon<0$ denote

 33.16.8 $\nu=1/(-\epsilon)^{1/2}(>0),$ β Defines: $\nu$: parameter (locally) Symbols: $\epsilon$: real parameter Referenced by: Β§33.16(v) Permalink: http://dlmf.nist.gov/33.16.E8 Encodings: TeX, pMML, png See also: Annotations for Β§33.16(iii), Β§33.16 and Ch.33
 33.16.9 $\displaystyle\zeta_{\ell}(\nu,r)$ $\displaystyle=W_{\nu,\ell+\frac{1}{2}}\left(2r/\nu\right),$ $\displaystyle\xi_{\ell}(\nu,r)$ $\displaystyle=\Re\left(e^{\mathrm{i}\pi\nu}W_{-\nu,\ell+\frac{1}{2}}\left(e^{% \mathrm{i}\pi}2r/\nu\right)\right),$ β Defines: $\zeta_{\ell}(\nu,r)$: function (locally) and $\xi_{\ell}(\nu,r)$: function (locally) Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\Re$: real part, $\ell$: nonnegative integer, $r$: real variable and $\nu$: parameter Referenced by: Β§33.16(v), Β§33.21(i) Permalink: http://dlmf.nist.gov/33.16.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for Β§33.16(iii), Β§33.16 and Ch.33

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

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

Alternatively, for $r<0$

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

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

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

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

Alternatively, for $r<0$

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

## Β§33.16(v) $s$ and $c$ in Terms of $W_{\kappa,\mu}\left(z\right)$ 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 $K(\nu,\ell)=\left(\nu^{2}\Gamma\left(\nu+\ell+1\right)\Gamma\left(\nu-\ell% \right)\right)^{-1/2}.$ β Defines: $K(\nu,\ell)$: factor (locally) Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\ell$: nonnegative integer and $\nu$: parameter Referenced by: Β§33.16(v) Permalink: http://dlmf.nist.gov/33.16.E16 Encodings: TeX, pMML, png See also: Annotations for Β§33.16(v), Β§33.16 and Ch.33

Then for $r>0$

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

Alternatively, for $r<0$

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