§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 $\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)=\left(\frac{1+x}{1-% x}\right)^{\mu/2}\mathop{\mathbf{F}\/}\nolimits\!\left(\nu+1,-\nu;1-\mu;\tfrac% {1}{2}-\tfrac{1}{2}x\right).$ Defines: $\mathop{\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: Ferrers function of the first kind Symbols: $\mathop{\mathbf{F}\/}\nolimits\!\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathop{\mathbf{F}\/}\nolimits\!\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z% }\right)$: Olver’s hypergeometric function, $x$: real variable, $\mu$: general order and $\nu$: general degree 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)

Ferrers Function of the Second Kind

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

Here and elsewhere in this chapter

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

is Olver’s hypergeometric function (§15.1).

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

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

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

equivalently,

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

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

Associated Legendre Function of the First Kind

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

Associated Legendre Function of the Second Kind

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

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

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

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

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

and

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

Like $\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$, but unlike $\mathop{Q^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$, $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ 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 $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ is due to Olver (1997b, pp. 170 and 178).

§14.3(iii) Alternative Hypergeometric Representations

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

where

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

For further hypergeometric representations of $\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ and $\mathop{Q^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ 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).

§14.3(iv) Relations to Other Functions

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

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

Compare also (18.11.1).