# §14.9 Connection Formulas

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

 14.9.1 $\frac{\pi\mathop{\sin\/}\nolimits\!\left(\mu\pi\right)}{2\mathop{\Gamma\/}% \nolimits\!\left(\nu-\mu+1\right)}\mathop{\mathsf{P}^{-\mu}_{\nu}\/}\nolimits% \!\left(x\right)=-\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}% \mathop{\mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)+\frac{\mathop{\cos% \/}\nolimits\!\left(\mu\pi\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1% \right)}\mathop{\mathsf{Q}^{-\mu}_{\nu}\/}\nolimits\!\left(x\right).$
 14.9.2 $\frac{2\mathop{\sin\/}\nolimits\!\left(\mu\pi\right)}{\pi\mathop{\Gamma\/}% \nolimits\!\left(\nu-\mu+1\right)}\mathop{\mathsf{Q}^{-\mu}_{\nu}\/}\nolimits% \!\left(x\right)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}% \mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)-\frac{\mathop{\cos% \/}\nolimits\!\left(\mu\pi\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1% \right)}\mathop{\mathsf{P}^{-\mu}_{\nu}\/}\nolimits\!\left(x\right),$
 14.9.3 $\mathop{\mathsf{P}^{-m}_{\nu}\/}\nolimits\!\left(x\right)=(-1)^{m}\frac{% \mathop{\Gamma\/}\nolimits\!\left(\nu-m+1\right)}{\mathop{\Gamma\/}\nolimits\!% \left(\nu+m+1\right)}\mathop{\mathsf{P}^{m}_{\nu}\/}\nolimits\!\left(x\right),$
 14.9.4 $\mathop{\mathsf{Q}^{-m}_{\nu}\/}\nolimits\!\left(x\right)=(-1)^{m}\frac{% \mathop{\Gamma\/}\nolimits\!\left(\nu-m+1\right)}{\mathop{\Gamma\/}\nolimits\!% \left(\nu+m+1\right)}\mathop{\mathsf{Q}^{m}_{\nu}\/}\nolimits\!\left(x\right),$ $\nu\neq m-1,m-2,\dots$.
 14.9.5 $\displaystyle\mathop{\mathsf{P}^{\mu}_{-\nu-1}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x\right),$ $\displaystyle\mathop{\mathsf{P}^{-\mu}_{-\nu-1}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{\mathsf{P}^{-\mu}_{\nu}\/}\nolimits\!\left(x\right),$ Symbols: $\mathop{\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: 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)
 14.9.6 $\pi\mathop{\cos\/}\nolimits\!\left(\nu\pi\right)\mathop{\cos\/}\nolimits\!% \left(\mu\pi\right)\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)=% \mathop{\sin\/}\nolimits\!\left((\nu+\mu)\pi\right)\mathop{\mathsf{Q}^{\mu}_{% \nu}\/}\nolimits\!\left(x\right)-\mathop{\sin\/}\nolimits\!\left((\nu-\mu)\pi% \right)\mathop{\mathsf{Q}^{\mu}_{-\nu-1}\/}\nolimits\!\left(x\right).$

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

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

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

 14.9.11 $\displaystyle\mathop{P^{-\mu}_{-\nu-1}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{P^{-\mu}_{\nu}\/}\nolimits\!\left(x\right),$ $\displaystyle\mathop{P^{\mu}_{-\nu-1}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\right),$ Symbols: $\mathop{P^{\NVar{\mu}}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 14.9.12 $\mathop{\cos\/}\nolimits\!\left(\nu\pi\right)\mathop{P^{-\mu}_{\nu}\/}% \nolimits\!\left(x\right)=-\frac{\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}% \nolimits\!\left(x\right)}{\mathop{\Gamma\/}\nolimits\!\left(\mu-\nu\right)}+% \frac{\mathop{\boldsymbol{Q}^{\mu}_{-\nu-1}\/}\nolimits\!\left(x\right)}{% \mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}.$
 14.9.13 $\mathop{P^{-m}_{\nu}\/}\nolimits\!\left(x\right)=\frac{\mathop{\Gamma\/}% \nolimits\!\left(\nu-m+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu+m+1% \right)}\mathop{P^{m}_{\nu}\/}\nolimits\!\left(x\right),$ $\nu\neq m-1,m-2,\dots$.
 14.9.14 $\mathop{\boldsymbol{Q}^{-\mu}_{\nu}\/}\nolimits\!\left(x\right)=\mathop{% \boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right),$ Symbols: $\mathop{\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 14.9.15 $\frac{2\mathop{\sin\/}\nolimits\!\left(\mu\pi\right)}{\pi}\mathop{\boldsymbol{% Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)=\frac{\mathop{P^{\mu}_{\nu}\/}% \nolimits\!\left(x\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}% -\frac{\mathop{P^{-\mu}_{\nu}\/}\nolimits\!\left(x\right)}{\mathop{\Gamma\/}% \nolimits\!\left(\nu-\mu+1\right)}.$

## §14.9(iv) Whipple’s Formula

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

Equivalently,

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