# §14.3 Definitions and Hypergeometric Representations

## §14.3(i) Interval $-1

The following are real-valued solutions of (14.2.2) when $\mu$, $\nu\in\mathbb{R}$ and $x\in(-1,1)$.

### Ferrers Function of the First Kind

 14.3.1 $\mathsf{P}^{\mu}_{\nu}\left(x\right)=\left(\frac{1+x}{1-x}\right)^{\mu/2}% \mathbf{F}\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right).$ ⓘ Defines: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$: Ferrers function of the first kind Symbols: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$: $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\mu$: general order, $\nu$: general degree and $x$: real variable $1 A&S Ref: 8.1.2 (modified) Referenced by: §14.11, §14.15(i), §14.3(i), §14.3(i), §15.9(iv), §15.9(iv) Permalink: http://dlmf.nist.gov/14.3.E1 Encodings: TeX, pMML, png See also: Annotations for §14.3(i), §14.3(i), §14.3 and Ch.14

### Ferrers Function of the Second Kind

 14.3.2 $\mathsf{Q}^{\mu}_{\nu}\left(x\right)=\frac{\pi}{2\sin\left(\mu\pi\right)}\left% (\cos\left(\mu\pi\right)\left(\frac{1+x}{1-x}\right)^{\mu/2}\mathbf{F}\left(% \nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)-\frac{\Gamma\left(\nu+\mu+1% \right)}{\Gamma\left(\nu-\mu+1\right)}\left(\frac{1-x}{1+x}\right)^{\mu/2}% \mathbf{F}\left(\nu+1,-\nu;1+\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)\right).$ ⓘ Defines: $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$: Ferrers function of the second kind Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$: $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $\sin\NVar{z}$: sine function, $\mu$: general order, $\nu$: general degree and $x$: real variable $1 Referenced by: §14.3(i) Permalink: http://dlmf.nist.gov/14.3.E2 Encodings: TeX, pMML, png See also: Annotations for §14.3(i), §14.3(i), §14.3 and Ch.14

Here and elsewhere in this chapter

 14.3.3 $\mathbf{F}\left(a,b;c;x\right)=\frac{1}{\Gamma\left(c\right)}F\left(a,b;c;x\right)$

is Olver’s hypergeometric function (§15.1).

$\mathsf{P}^{\mu}_{\nu}\left(x\right)$ exists for all values of $\mu$ and $\nu$. $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ is undefined when $\mu+\nu=-1,-2,-3,\dots$.

When $\mu=m=0,1,2,\dotsc$, (14.3.1) reduces to

 14.3.4 $\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu+m+1\right)}{2% ^{m}\Gamma\left(\nu-m+1\right)}\left(1-x^{2}\right)^{m/2}\mathbf{F}\left(\nu+m% +1,m-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right);$

equivalently,

 14.3.5 $\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\frac{\Gamma\left(\nu+m+1\right)}{% \Gamma\left(\nu-m+1\right)}\left(\frac{1-x}{1+x}\right)^{m/2}\mathbf{F}\left(% \nu+1,-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right).$

When $\mu=m$ ($\in\mathbb{Z}$) (14.3.2) is replaced by its limiting value; see Hobson (1931, §132) for details. See also (14.3.12)–(14.3.14) for this case.

## §14.3(ii) Interval $1

The following are solutions of (14.2.2) when $\mu$, $\nu\in\mathbb{R}$ and $x>1$.

### Associated Legendre Function of the First Kind

 14.3.6 $P^{\mu}_{\nu}\left(x\right)=\left(\frac{x+1}{x-1}\right)^{\mu/2}\mathbf{F}% \left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right).$

### Associated Legendre Function of the Second Kind

 14.3.7 $Q^{\mu}_{\nu}\left(x\right)=e^{\mu\pi i}\frac{\pi^{1/2}\Gamma\left(\nu+\mu+1% \right)\left(x^{2}-1\right)^{\mu/2}}{2^{\nu+1}x^{\nu+\mu+1}}\mathbf{F}\left(% \tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1,\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac{1}{2}% ;\nu+\tfrac{3}{2};\frac{1}{x^{2}}\right),$ $\mu+\nu\neq-1,-2,-3,\dots$.

When $\mu=m=1,2,3,\dots$, (14.3.6) reduces to

 14.3.8 $P^{m}_{\nu}\left(x\right)=\frac{\Gamma\left(\nu+m+1\right)}{2^{m}\Gamma\left(% \nu-m+1\right)}\left(x^{2}-1\right)^{m/2}\mathbf{F}\left(\nu+m+1,m-\nu;m+1;% \tfrac{1}{2}-\tfrac{1}{2}x\right).$

As standard solutions of (14.2.2) we take the pair $P^{-\mu}_{\nu}\left(x\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$, where

 14.3.9 $P^{-\mu}_{\nu}\left(x\right)=\left(\frac{x-1}{x+1}\right)^{\mu/2}\mathbf{F}% \left(\nu+1,-\nu;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right),$

and

 14.3.10 $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=e^{-\mu\pi i}\frac{Q^{\mu}_{\nu}\left% (x\right)}{\Gamma\left(\nu+\mu+1\right)}.$

Like $P^{\mu}_{\nu}\left(x\right)$, but unlike $Q^{\mu}_{\nu}\left(x\right)$, $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ is real-valued when $\nu$, $\mu\in\mathbb{R}$ and $x\in(1,\infty)$, and is defined for all values of $\nu$ and $\mu$. The notation $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ is due to Olver (1997b, pp. 170 and 178).

## §14.3(iii) Alternative Hypergeometric Representations

 14.3.11 $\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)$ $\displaystyle=\cos\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{1}(\nu,\mu,x)+\sin% \left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{2}(\nu,\mu,x),$ 14.3.12 $\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ $\displaystyle=-\tfrac{1}{2}\pi\sin\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{1}(% \nu,\mu,x)+\tfrac{1}{2}\pi\cos\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{2}(\nu,% \mu,x),$

where

 14.3.13 $\displaystyle w_{1}(\nu,\mu,x)$ $\displaystyle=\frac{2^{\mu}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{% 2}\right)}{\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1\right)}\left(1-x^{2}% \right)^{-\mu/2}\mathbf{F}\left(-\tfrac{1}{2}\nu-\tfrac{1}{2}\mu,\tfrac{1}{2}% \nu-\tfrac{1}{2}\mu+\tfrac{1}{2};\tfrac{1}{2};x^{2}\right),$ 14.3.14 $\displaystyle w_{2}(\nu,\mu,x)$ $\displaystyle=\frac{2^{\mu}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)}% {\Gamma\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}\right)}x\left(1-x^{2}% \right)^{-\mu/2}\mathbf{F}\left(\tfrac{1}{2}-\tfrac{1}{2}\nu-\tfrac{1}{2}\mu,% \tfrac{1}{2}\nu-\tfrac{1}{2}\mu+1;\tfrac{3}{2};x^{2}\right).$
 14.3.15 $P^{-\mu}_{\nu}\left(x\right)=2^{-\mu}\left(x^{2}-1\right)^{\mu/2}\mathbf{F}% \left(\mu-\nu,\nu+\mu+1;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right),$
 14.3.16 $\cos\left(\nu\pi\right)P^{-\mu}_{\nu}\left(x\right)=\frac{2^{\nu}\pi^{1/2}x^{% \nu-\mu}\left(x^{2}-1\right)^{\mu/2}}{\Gamma\left(\nu+\mu+1\right)}\mathbf{F}% \left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1% }{2};\tfrac{1}{2}-\nu;\frac{1}{x^{2}}\right)-\frac{\pi^{1/2}\left(x^{2}-1% \right)^{\mu/2}}{2^{\nu+1}\Gamma\left(\mu-\nu\right)x^{\nu+\mu+1}}\mathbf{F}% \left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1,\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+\tfrac% {1}{2};\nu+\tfrac{3}{2};\frac{1}{x^{2}}\right),$
 14.3.17 $P^{-\mu}_{\nu}\left(x\right)=\frac{\pi\left(x^{2}-1\right)^{\mu/2}}{2^{\mu}}% \left(\frac{\mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu,\frac{1}{2}\nu+\frac% {1}{2}\mu+\frac{1}{2};\frac{1}{2};x^{2}\right)}{\Gamma\left(\frac{1}{2}\mu-% \frac{1}{2}\nu+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1% \right)}-\frac{x\mathbf{F}\left(\frac{1}{2}\mu-\frac{1}{2}\nu+\frac{1}{2},% \frac{1}{2}\nu+\frac{1}{2}\mu+1;\frac{3}{2};x^{2}\right)}{\Gamma\left(\frac{1}% {2}\mu-\frac{1}{2}\nu\right)\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}% {2}\right)}\right),$
 14.3.18 $\displaystyle P^{-\mu}_{\nu}\left(x\right)$ $\displaystyle=2^{-\mu}x^{\nu-\mu}\left(x^{2}-1\right)^{\mu/2}\mathbf{F}\left(% \tfrac{1}{2}\mu-\tfrac{1}{2}\nu,\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2};% \mu+1;1-\frac{1}{x^{2}}\right),$ 14.3.19 $\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ $\displaystyle=\frac{2^{\nu}\Gamma\left(\nu+1\right)(x+1)^{\mu/2}}{(x-1)^{(\mu/% 2)+\nu+1}}\mathbf{F}\left(\nu+1,\nu+\mu+1;2\nu+2;\frac{2}{1-x}\right),$
 14.3.20 $\frac{2\sin\left(\mu\pi\right)}{\pi}\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=% \frac{(x+1)^{\mu/2}}{\Gamma\left(\nu+\mu+1\right)(x-1)^{\mu/2}}\mathbf{F}\left% (\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)-\frac{(x-1)^{\mu/2}}{% \Gamma\left(\nu-\mu+1\right)(x+1)^{\mu/2}}\mathbf{F}\left(\nu+1,-\nu;\mu+1;% \tfrac{1}{2}-\tfrac{1}{2}x\right).$

For further hypergeometric representations of $P^{\mu}_{\nu}\left(x\right)$ and $Q^{\mu}_{\nu}\left(x\right)$ see Erdélyi et al. (1953a, pp. 123–139), Andrews et al. (1999, §3.1), Magnus et al. (1966, pp. 153–163), and §15.8(iii). For further hypergeometric representations of $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ see Cohl et al. (2021).

## §14.3(iv) Relations to Other Functions

In terms of the Gegenbauer function $C^{(\beta)}_{\alpha}\left(x\right)$ and the Jacobi function $\phi^{(\alpha,\beta)}_{\lambda}\left(t\right)$ (§§15.9(iii), 15.9(ii)):

 14.3.21 $\displaystyle\mathsf{P}^{\mu}_{\nu}\left(x\right)$ $\displaystyle=\frac{2^{\mu}\Gamma\left(1-2\mu\right)\Gamma\left(\nu+\mu+1% \right)}{\Gamma\left(\nu-\mu+1\right)\Gamma\left(1-\mu\right)\left(1-x^{2}% \right)^{\mu/2}}C^{(\frac{1}{2}-\mu)}_{\nu+\mu}\left(x\right).$ 14.3.22 $\displaystyle P^{\mu}_{\nu}\left(x\right)$ $\displaystyle=\frac{2^{\mu}\Gamma\left(1-2\mu\right)\Gamma\left(\nu+\mu+1% \right)}{\Gamma\left(\nu-\mu+1\right)\Gamma\left(1-\mu\right)\left(x^{2}-1% \right)^{\mu/2}}C^{(\frac{1}{2}-\mu)}_{\nu+\mu}\left(x\right).$ 14.3.23 $\displaystyle P^{\mu}_{\nu}\left(x\right)$ $\displaystyle=\frac{1}{\Gamma\left(1-\mu\right)}\left(\frac{x+1}{x-1}\right)^{% \mu/2}\phi^{(-\mu,\mu)}_{-\mathrm{i}(2\nu+1)}\left(\operatorname{arcsinh}\left% ((\tfrac{1}{2}x-\tfrac{1}{2})^{\ifrac{1}{2}}\right)\right).$

Compare also (18.11.1). From (15.9.15) it follows that $1-2\mu=0,-1,-2,\dots$ and $\nu+\mu+1=0,-1,-2,\dots$ are removable singularities of the right-hand sides of (14.3.21) and (14.3.22).