# Ferrers functions

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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). …
##### 2: 14.3 Definitions and Hypergeometric Representations
###### FerrersFunction of the First Kind
14.3.1 $\mathsf{P}^{\mu}_{\nu}\left(x\right)=\left(\frac{1+x}{1-x}\right)^{\mu/2}% \mathbf{F}\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right).$
###### FerrersFunction of the Second Kind
14.3.2 $\mathsf{Q}^{\mu}_{\nu}\left(x\right)=\frac{\pi}{2\sin\left(\mu\pi\right)}\left% (\cos\left(\mu\pi\right)\left(\frac{1+x}{1-x}\right)^{\mu/2}\mathbf{F}\left(% \nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)-\frac{\Gamma\left(\nu+\mu+1% \right)}{\Gamma\left(\nu-\mu+1\right)}\left(\frac{1-x}{1+x}\right)^{\mu/2}% \mathbf{F}\left(\nu+1,-\nu;1+\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)\right).$
##### 3: 14.20 Conical (or Mehler) Functions
For $-1 and $\tau>0$, a numerically satisfactory pair of real conical functions is $\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$ and $\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(-x\right)$. …
14.20.4 $\mathscr{W}\left\{\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right% ),\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(-x\right)\right\}=\frac% {2}{|\Gamma\left(\mu+\frac{1}{2}+\mathrm{i}\tau\right)|^{2}(1-x^{2})}.$
14.20.7 $\widehat{\mathsf{Q}}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)\sim\tfrac{1}{2}% \Gamma\left(\mu\right)\left(\frac{2}{1-x}\right)^{\mu/2},$
14.20.22 $\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{\beta\exp\left(\mu% \beta\operatorname{arctan}\beta\right)}{\Gamma\left(\mu+1\right)\left(1+\beta^% {2}\right)^{\mu/2}}\frac{e^{-\mu\rho}}{\left(1+\beta^{2}-x^{2}\beta^{2}\right)% ^{1/4}}\left(1+O\left(\frac{1}{\mu}\right)\right),$
##### 4: 14.2 Differential Equations
###### §14.2(ii) Associated Legendre Equation
Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations $\mathsf{P}^{0}_{\nu}\left(x\right)=\mathsf{P}_{\nu}\left(x\right)$, $\mathsf{Q}^{0}_{\nu}\left(x\right)=\mathsf{Q}_{\nu}\left(x\right)$, $P^{0}_{\nu}\left(x\right)=P_{\nu}\left(x\right)$, $Q^{0}_{\nu}\left(x\right)=Q_{\nu}\left(x\right)$, $\boldsymbol{Q}^{0}_{\nu}\left(x\right)=\boldsymbol{Q}_{\nu}\left(x\right)=Q_{% \nu}\left(x\right)/\Gamma\left(\nu+1\right)$. … Unless stated otherwise in §§14.214.20 it is assumed that the arguments of the functions $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ and $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ lie in the interval $(-1,1)$, and the arguments of the functions $P^{\mu}_{\nu}\left(x\right)$, $Q^{\mu}_{\nu}\left(x\right)$, and $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ lie in the interval $(1,\infty)$. …
###### §14.2(iv) Wronskians and Cross-Products
14.2.5 $\mathsf{P}^{\mu}_{\nu+1}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)-% \mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu+1}\left(x\right)=% \frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+2\right)},$
##### 9: 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 $\left(1-x^{2}\right)\frac{\mathrm{d}\mathsf{P}^{\mu}_{\nu}\left(x\right)}{% \mathrm{d}x}={(\mu-\nu-1)\mathsf{P}^{\mu}_{\nu+1}\left(x\right)+(\nu+1)x% \mathsf{P}^{\mu}_{\nu}\left(x\right)},$
##### 10: 14.4 Graphics
###### §14.4(ii) FerrersFunctions: 3D Surfaces Figure 14.4.15: P 0 - μ ⁡ ( x ) , 0 ≤ μ ≤ 10 , - 1 < x < 1 . Magnify 3D Help Figure 14.4.16: Q 0 μ ⁡ ( x ) , 0 ≤ μ ≤ 6.2 , - 1 < x < 1 . Magnify 3D Help