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14 Legendre and Related FunctionsReal Arguments

§14.3 Definitions and Hypergeometric Representations

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§14.3(i) Interval -1<x<1

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Notes:
See Olver (1997b, pp. 159, 186). The version of (14.3.4) given in Hobson (1931, p. 386) has an error. For (14.3.5) use (14.3.1) and (14.9.3).
Keywords:
Ferrers functions, hypergeometric function
Referenced by:
‣ §10.22(iv), §10.43(iv), §14.20(i), §14.23, §14.32, §15.9(iv), ‣ §18.11(i), ‣ §18.17(iii), §29.8, §30.4(i), §30.5, §30.8(i)
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http://dlmf.nist.gov/14.3.i

The following are real-valued solutions of (14.2.2) when \mu, \nu\in\Real and x\in(-1,1).

Ferrers Function of the First Kind

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Keywords:
Ferrers functions
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).
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Defines:
\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right): Ferrers function of the first kind
Symbols:
\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;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)
Permalink:
http://dlmf.nist.gov/14.3.E1
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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).

Here and elsewhere in this chapter

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

equivalently,

When \mu=m (\in\Integer) (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<x<\infty

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Notes:
See Olver (1997b, p. 170). For (14.3.8) see Olver (1997b, p. 159, Eq. (9.05)).
Keywords:
associated Legendre functions, hypergeometric function
Referenced by:
‣ §10.22(iv), ‣ §10.22(iv), ‣ §18.17(iii), ‣ §18.30, ‣ §30.6
Permalink:
http://dlmf.nist.gov/14.3.ii

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).
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Defines:
\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right): associated Legendre function of the first kind
Symbols:
\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right): Olver’s hypergeometric function, x: real variable, z: complex variable, \mu: general order and \nu: general degree
Referenced by:
§14.21(i), §14.21(iii), ‣ §14.3(ii), §14.3(iv), §15.9(iv)
Permalink:
http://dlmf.nist.gov/14.3.E6
Encodings:
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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.
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Defines:
\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right): associated Legendre function of the second kind
Symbols:
\mathop{\Gamma\/}\nolimits\!\left(z\right): gamma function, e: base of exponential function, \mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right): Olver’s hypergeometric function, x: real variable, z: complex variable, \mu: general order and \nu: general degree
A&S Ref:
8.1.3 (modified)
Permalink:
http://dlmf.nist.gov/14.3.E7
Encodings:
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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

and

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\Real 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

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Notes:
For (14.3.11) and (14.3.12) see Olver (1997b, p. 187). For (14.3.16)–(14.3.20) see Erdélyi et al. (1953a, pp. 123–139).
Keywords:
Ferrers functions, associated Legendre functions, hypergeometric function
Referenced by:
§14.32, §15.9(iv)
Permalink:
http://dlmf.nist.gov/14.3.iii

where

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).

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