14 Legendre and Related Functions Real Arguments 14.12 Integral Representations 14.14 Continued Fractions

§14.13 Trigonometric Expansions

ⓘ

Keywords:

Ferrers functions, trigonometric expansions

Notes:

The expansions (14.13.1) and (14.13.2) are derived in §3.5 of Erdélyi et al. (1953a), but without any statement of conditions on

\nu

and

\mu

. To obtain these conditions we see that by combination of (14.13.1) and (14.13.2),

\pm\frac{1}{2}\pi i\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)+\mathsf{Q}^{% \mu}_{\nu}\left(\cos\theta\right)=\pi^{\frac{1}{2}}\Gamma\left(\nu+\mu+1\right% )(2\sin\theta)^{\mu}e^{\pm(\nu+\mu+1)i\theta}\*\mathbf{F}\left(\nu+\mu+1,\mu+% \frac{1}{2};\nu+\frac{3}{2};e^{\pm 2i\theta}\right).

We then apply the results given in the last paragraph of §15.2(i), and subsequently take real and imaginary parts in the case of conditional convergence.

Referenced by:

Erratum (V1.0.1) for Clarifications

Permalink:

http://dlmf.nist.gov/14.13

Correction (effective with 1.1.3):

The conditions for convergence of the Fourier expansions (14.13.1), (14.13.2) have been corrected. Originally it was stated that they converge “…conditionally when

\nu

is real and

0\leq\mu<\frac{1}{2}

.” It has been corrected to read “If

0\leq\Re\mu<\frac{1}{2}

then they converge, but, if

\theta\not=\frac{1}{2}\pi

, they do not converge absolutely.”

Suggested 2021-09-08 by Hans Volkmer

Clarification (effective with 1.0.1):

The conditions for convergence of the expansions in this subsection were clarified.

See also:

Annotations for Ch.14

When $0<\theta<\pi$ , and $\nu+\mu$ is not a negative integer,

14.13.1	$\displaystyle\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)$	$\displaystyle=\frac{2^{\mu+1}(\sin\theta)^{\mu}}{\pi^{1/2}}\\sum_{k=0}^{% \infty}\frac{\Gamma\left(\nu+\mu+k+1\right)}{\Gamma\left(\nu+k+\frac{3}{2}% \right)}\frac{{\left(\mu+\frac{1}{2}\right)_{k}}}{k!}\\sin\left((\nu+\mu+2k+1% )\theta\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $\mu$ : general order, $\nu$ : general degree and $0<\theta<\pi$ : variable Source: Volkmer (2021) A&S Ref: 8.7.1 Referenced by: §14.13, Erratum (V1.1.3) for Equations (14.13.1), (14.13.2) Permalink: http://dlmf.nist.gov/14.13.E1 Encodings: TeX, pMML, png See also: Annotations for §14.13 and Ch.14
14.13.2	$\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)$	$\displaystyle=\pi^{1/2}2^{\mu}(\sin\theta)^{\mu}\\sum_{k=0}^{\infty}\frac{% \Gamma\left(\nu+\mu+k+1\right)}{\Gamma\left(\nu+k+\frac{3}{2}\right)}\frac{{% \left(\mu+\frac{1}{2}\right)_{k}}}{k!}\\cos\left((\nu+\mu+2k+1)\theta\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : sine function, $\mu$ : general order, $\nu$ : general degree and $0<\theta<\pi$ : variable Source: Cohl et al. (2021, Theorem 6.2) A&S Ref: 8.7.2 Referenced by: §14.13, Erratum (V1.1.3) for Equations (14.13.1), (14.13.2) Permalink: http://dlmf.nist.gov/14.13.E2 Encodings: TeX, pMML, png See also: Annotations for §14.13 and Ch.14

These Fourier series converge absolutely when $\Re\mu<0$ . If $0\leq\Re\mu<\frac{1}{2}$ then they converge, but, if $\theta\not=\frac{1}{2}\pi$ , they do not converge absolutely.

In particular,

14.13.3	$\displaystyle\mathsf{P}_{n}\left(\cos\theta\right)$	$\displaystyle=\frac{2^{2n+2}(n!)^{2}}{\pi(2n+1)!}\\sum_{k=0}^{\infty}\frac{1% \cdot 3\cdots(2k-1)}{k!}\\frac{(n+1)(n+2)\cdots(n+k)}{(2n+3)(2n+5)\cdots(2n+2% k+1)}\*\sin\left((n+2k+1)\theta\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\sin\NVar{z}$ : sine function, $n$ : nonnegative integer and $0<\theta<\pi$ : variable A&S Ref: 8.7.3 Permalink: http://dlmf.nist.gov/14.13.E3 Encodings: TeX, pMML, png See also: Annotations for §14.13 and Ch.14
14.13.4	$\displaystyle\mathsf{Q}_{n}\left(\cos\theta\right)$	$\displaystyle=\frac{2^{2n+1}(n!)^{2}}{(2n+1)!}\\sum_{k=0}^{\infty}\frac{1% \cdot 3\cdots(2k-1)}{k!}\\frac{(n+1)(n+2)\cdots(n+k)}{(2n+3)(2n+5)\cdots(2n+2% k+1)}\*\cos\left((n+2k+1)\theta\right),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, $n$ : nonnegative integer and $0<\theta<\pi$ : variable A&S Ref: 8.7.4 Permalink: http://dlmf.nist.gov/14.13.E4 Encodings: TeX, pMML, png See also: Annotations for §14.13 and Ch.14

with conditional convergence for each.

For other trigonometric expansions see Erdélyi et al. (1953a, pp. 146–147).

14.12 Integral Representations 14.14 Continued Fractions

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§14.13 Trigonometric Expansions

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