relation to associated Legendre functions

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1: 14.3 Definitions and Hypergeometric Representations

§14.3 Definitions and Hypergeometric Representations

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§14.3(ii) Interval

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§14.3(iii) Alternative Hypergeometric Representations

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14.3.14 $$w_2(\nu ,\mu ,x)=\frac{2^{\mu }\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +1\right)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)}x{\left(1-x^2\right)}^{-\mu /2}\mathbf{F}(\frac{1}{2}-\frac{1}{2}\nu -\frac{1}{2}\mu ,\frac{1}{2}\nu -\frac{1}{2}\mu +1;\frac{3}{2};x^2).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathbf{F}(a,b;c;z)$ or $\mathbf{F}(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2\mathbf{F}_1(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $\mu$: general order, $\nu$: general degree and $w_2$: function $w_1$,

Referenced by:

§10.59, §14.3(i), §14.5(i)

Permalink:

http://dlmf.nist.gov/14.3.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iii), §14.3 and Ch.14

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§14.3(iv) Relations to Other Functions

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2: 14.7 Integer Degree and Order

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§14.7(i) $\mu =0$

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3: 14.5 Special Values

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§14.5(v) $\mu =0$, $\nu =\pm \frac{1}{2}$

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4: 15.9 Relations to Other Functions

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§15.9(iv) Associated Legendre Functions; Ferrers Functions

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5: 30.6 Functions of Complex Argument

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Relations to Associated Legendre Functions

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6: 14.2 Differential Equations

… ►Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations ${\mathsf{P}}_{\nu }^0\left(x\right)={\mathsf{P}}_{\nu }\left(x\right)$, ${\mathsf{Q}}_{\nu }^0\left(x\right)={\mathsf{Q}}_{\nu }\left(x\right)$, $P_{\nu }^0\left(x\right)=P_{\nu }\left(x\right)$, $Q_{\nu }^0\left(x\right)=Q_{\nu }\left(x\right)$, ${\mathit{Q}}_{\nu }^0\left(x\right)={\mathit{Q}}_{\nu }\left(x\right)=Q_{\nu }\left(x\right)/\mathrm{\Gamma }\left(\nu +1\right)$. …

7: 14.21 Definitions and Basic Properties

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§14.21(i) Associated Legendre Equation

… ►Standard solutions: the associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $P_{\nu }^{-\mu }\left(z\right)$, ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$, and ${\mathit{Q}}_{-\nu -1}^{\mu }\left(z\right)$. … ►

§14.21(ii) Numerically Satisfactory Solutions

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§14.21(iii) Properties

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8: 14 Legendre and Related Functions

Chapter 14 Legendre and Related Functions

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9: 16.18 Special Cases

§16.18 Special Cases

… ►This is a consequence of the following relations: …As a corollary, special cases of the ${}_{1}F_1$ and ${}_{2}F_1$ functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer $G$-function. Representations of special functions in terms of the Meijer $G$-function are given in Erdélyi et al. (1953a, §5.6), Luke (1969a, §§6.4–6.5), and Mathai (1993, §3.10).

10: 14.32 Methods of Computation

§14.32 Methods of Computation

►Essentially the same comments that are made in §15.19 concerning the computation of hypergeometric functions apply to the functions described in the present chapter. In particular, for small or moderate values of the parameters $\mu$ and $\nu$ the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. In other cases recurrence relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967). … ►

For the computation of conical functions see Gil et al. (2009, 2012), and Dunster (2014).