# Ferrers function

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##### 1: 14.1 Special Notation
###### §14.1 Special Notation
The main functions treated in this chapter are the Legendre functions $\mathsf{P}_{\nu}\left(x\right)$, $\mathsf{Q}_{\nu}\left(x\right)$, $P_{\nu}\left(z\right)$, $Q_{\nu}\left(z\right)$; Ferrers functions $\mathsf{P}^{\mu}_{\nu}\left(x\right)$, $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P^{\mu}_{\nu}\left(z\right)$, $Q^{\mu}_{\nu}\left(z\right)$, $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$; conical functions $\mathsf{P}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $\mathsf{Q}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $\widehat{\mathsf{Q}}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $P^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $Q^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$ (also known as Mehler functions). …
##### 6: 14.10 Recurrence Relations and Derivatives
###### §14.10 Recurrence Relations and Derivatives
14.10.1 ${\mathsf{P}^{\mu+2}_{\nu}\left(x\right)+2(\mu+1)x\left(1-x^{2}\right)^{-1/2}% \mathsf{P}^{\mu+1}_{\nu}\left(x\right)}+(\nu-\mu)(\nu+\mu+1)\mathsf{P}^{\mu}_{% \nu}\left(x\right)=0,$
14.10.2 ${\left(1-x^{2}\right)^{1/2}\mathsf{P}^{\mu+1}_{\nu}\left(x\right)-(\nu-\mu+1)% \mathsf{P}^{\mu}_{\nu+1}\left(x\right)}+(\nu+\mu+1)x\mathsf{P}^{\mu}_{\nu}% \left(x\right)=0,$
14.10.3 ${(\nu-\mu+2)\mathsf{P}^{\mu}_{\nu+2}\left(x\right)-(2\nu+3)x\mathsf{P}^{\mu}_{% \nu+1}\left(x\right)}+(\nu+\mu+1)\mathsf{P}^{\mu}_{\nu}\left(x\right)=0,$
14.10.4 $\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{% \mathrm{d}x}={(\mu-\nu-1)\mathsf{P}^{\mu}_{\nu+1}\left(x\right)+(\nu+1)x% \mathsf{P}^{\mu}_{\nu}\left(x\right)},$
##### 8: 14.18 Sums
###### §14.18 Sums
14.18.3 $\mathsf{Q}_{\nu}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2% }\cos\phi\right)=\mathsf{P}_{\nu}\left(\cos\theta_{1}\right)\mathsf{Q}_{\nu}% \left(\cos\theta_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\mathsf{P}^{-m}_{\nu}% \left(\cos\theta_{1}\right)\mathsf{Q}^{m}_{\nu}\left(\cos\theta_{2}\right)\cos% \left(m\phi\right).$
The formulas are also valid with the Ferrers functions as in §14.3(i) with $\mu=0$. …
###### §14.9(i) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(x\right)$, $\mathsf{P}^{\pm\mu}_{-\nu-1}\left(x\right)$, $\mathsf{Q}^{\pm\mu}_{\nu}\left(x\right)$, $\mathsf{Q}^{\mu}_{-\nu-1}\left(x\right)$
14.9.4 $\mathsf{Q}^{-m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{% \Gamma\left(\nu+m+1\right)}\mathsf{Q}^{m}_{\nu}\left(x\right),$ $\nu\neq m-1,m-2,\dots$.
###### §14.9(ii) Connections Between $\mathsf{P}^{\pm\mu}_{\nu}\left(\pm x\right)$, $\mathsf{Q}^{-\mu}_{\nu}\left(\pm x\right)$, $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$
14.9.9 $\frac{2}{\Gamma\left(\nu+\mu+1\right)\Gamma\left(\mu-\nu\right)}\mathsf{Q}^{% \mu}_{\nu}\left(x\right)=-\cos\left(\nu\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(% x\right)+\cos\left(\mu\pi\right)\mathsf{P}^{-\mu}_{\nu}\left(-x\right),$
14.6.1 $\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{% \mathrm{d}}^{m}\mathsf{P}_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}},$
14.6.2 $\mathsf{Q}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(1-x^{2}\right)^{m/2}\frac{{% \mathrm{d}}^{m}\mathsf{Q}_{\nu}\left(x\right)}{{\mathrm{d}x}^{m}}.$