# §14.9 Connection Formulas

## §14.9(i) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(x\right)$, $\mathsf{P}^{\pm\mu}_{-\nu-1}\left(x\right)$, $\mathsf{Q}^{\pm\mu}_{\nu}\left(x\right)$, $\mathsf{Q}^{\mu}_{-\nu-1}\left(x\right)$

 14.9.1 $\frac{\pi\sin\left(\mu\pi\right)}{2\Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-% \mu}_{\nu}\left(x\right)=-\frac{1}{\Gamma\left(\nu+\mu+1\right)}\mathsf{Q}^{% \mu}_{\nu}\left(x\right)+\frac{\cos\left(\mu\pi\right)}{\Gamma\left(\nu-\mu+1% \right)}\mathsf{Q}^{-\mu}_{\nu}\left(x\right).$
 14.9.2 $\frac{2\sin\left(\mu\pi\right)}{\pi\Gamma\left(\nu-\mu+1\right)}\mathsf{Q}^{-% \mu}_{\nu}\left(x\right)=\frac{1}{\Gamma\left(\nu+\mu+1\right)}\mathsf{P}^{\mu% }_{\nu}\left(x\right)-\frac{\cos\left(\mu\pi\right)}{\Gamma\left(\nu-\mu+1% \right)}\mathsf{P}^{-\mu}_{\nu}\left(x\right),$
 14.9.3 $\mathsf{P}^{-m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{% \Gamma\left(\nu+m+1\right)}\mathsf{P}^{m}_{\nu}\left(x\right),$
 14.9.4 $\mathsf{Q}^{-m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{% \Gamma\left(\nu+m+1\right)}\mathsf{Q}^{m}_{\nu}\left(x\right),$ $\nu\neq m-1,m-2,\dots$.
 14.9.5 $\displaystyle\mathsf{P}^{\mu}_{-\nu-1}\left(x\right)$ $\displaystyle=\mathsf{P}^{\mu}_{\nu}\left(x\right),$ $\displaystyle\mathsf{P}^{-\mu}_{-\nu-1}\left(x\right)$ $\displaystyle=\mathsf{P}^{-\mu}_{\nu}\left(x\right),$ ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$: Ferrers function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree A&S Ref: 8.2.1 Referenced by: §14.16(i) Permalink: http://dlmf.nist.gov/14.9.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 14.9(i), 14.9 and 14
 14.9.6 $\pi\cos\left(\nu\pi\right)\cos\left(\mu\pi\right)\mathsf{P}^{\mu}_{\nu}\left(x% \right)=\sin\left((\nu+\mu)\pi\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)-\sin% \left((\nu-\mu)\pi\right)\mathsf{Q}^{\mu}_{-\nu-1}\left(x\right).$

## §14.9(ii) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(\pm x\right)$, $\mathsf{Q}^{-\mu}_{\nu}\left(\pm x\right)$, $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$

 14.9.7 $\frac{\sin\left((\nu-\mu)\pi\right)}{\Gamma\left(\nu+\mu+1\right)}\mathsf{P}^{% \mu}_{\nu}\left(x\right)=\frac{\sin\left(\nu\pi\right)}{\Gamma\left(\nu-\mu+1% \right)}\mathsf{P}^{-\mu}_{\nu}\left(x\right)-\frac{\sin\left(\mu\pi\right)}{% \Gamma\left(\nu-\mu+1\right)}\mathsf{P}^{-\mu}_{\nu}\left(-x\right),$
 14.9.8 $\tfrac{1}{2}\pi\sin\left((\nu-\mu)\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(x% \right)=-\cos\left((\nu-\mu)\pi\right)\mathsf{Q}^{-\mu}_{\nu}\left(x\right)-% \mathsf{Q}^{-\mu}_{\nu}\left(-x\right),$
 14.9.9 $\frac{2}{\Gamma\left(\nu+\mu+1\right)\Gamma\left(\mu-\nu\right)}\mathsf{Q}^{% \mu}_{\nu}\left(x\right)=-\cos\left(\nu\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(% x\right)+\cos\left(\mu\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(-x\right),$
 14.9.10 $(2/\pi)\sin\left((\nu-\mu)\pi\right)\mathsf{Q}^{-\mu}_{\nu}\left(x\right)=\cos% \left((\nu-\mu)\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(x\right)-\mathsf{P}^{-% \mu}_{\nu}\left(-x\right).$

## §14.9(iii) Connections Between $P^{\pm\mu}_{\nu}\left(x\right)$, $P^{\pm\mu}_{-\nu-1}\left(x\right)$, $\boldsymbol{Q}^{\pm\mu}_{\nu}\left(x\right)$, $\boldsymbol{Q}^{\mu}_{-\nu-1}\left(x\right)$

 14.9.11 $\displaystyle P^{-\mu}_{-\nu-1}\left(x\right)$ $\displaystyle=P^{-\mu}_{\nu}\left(x\right),$ $\displaystyle P^{\mu}_{-\nu-1}\left(x\right)$ $\displaystyle=P^{\mu}_{\nu}\left(x\right),$ ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$: associated Legendre function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree A&S Ref: 8.2.1 Referenced by: §14.16(i), §14.19(v), §14.21(iii) Permalink: http://dlmf.nist.gov/14.9.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 14.9(iii), 14.9 and 14
 14.9.12 $\cos\left(\nu\pi\right)P^{-\mu}_{\nu}\left(x\right)=-\frac{\boldsymbol{Q}^{\mu% }_{\nu}\left(x\right)}{\Gamma\left(\mu-\nu\right)}+\frac{\boldsymbol{Q}^{\mu}_% {-\nu-1}\left(x\right)}{\Gamma\left(\nu+\mu+1\right)}.$
 14.9.13 $P^{-m}_{\nu}\left(x\right)=\frac{\Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m% +1\right)}P^{m}_{\nu}\left(x\right),$ $\nu\neq m-1,m-2,\dots$.
 14.9.14 $\boldsymbol{Q}^{-\mu}_{\nu}\left(x\right)=\boldsymbol{Q}^{\mu}_{\nu}\left(x% \right),$ ⓘ Symbols: $\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$: Olver’s associated Legendre function, $x$: real variable, $\mu$: general order and $\nu$: general degree A&S Ref: 8.2.6 (modified) Referenced by: §14.23, §14.9(iv) Permalink: http://dlmf.nist.gov/14.9.E14 Encodings: TeX, pMML, png See also: Annotations for 14.9(iii), 14.9 and 14
 14.9.15 $\frac{2\sin\left(\mu\pi\right)}{\pi}\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=% \frac{P^{\mu}_{\nu}\left(x\right)}{\Gamma\left(\nu+\mu+1\right)}-\frac{P^{-\mu% }_{\nu}\left(x\right)}{\Gamma\left(\nu-\mu+1\right)}.$

## §14.9(iv) Whipple’s Formula

 14.9.16 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\left(\tfrac{1}{2}\pi\right)^{1/2}% \left(x^{2}-1\right)^{-1/4}\*P^{-\nu-(1/2)}_{-\mu-(1/2)}\left(x\left(x^{2}-1% \right)^{-1/2}\right).$

Equivalently,

 14.9.17 $P^{\mu}_{\nu}\left(x\right)=(2/\pi)^{1/2}\left(x^{2}-1\right)^{-1/4}\*% \boldsymbol{Q}^{\nu+(1/2)}_{-\mu-(1/2)}\left(x\left(x^{2}-1\right)^{-1/2}% \right).$