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

§14.13 Trigonometric Expansions

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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 $\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)=\frac{2^{\mu+1}(\sin\theta)^{\mu% }}{\pi^{1/2}}\*\sum_{k=0}^{\infty}\frac{\Gamma\left(\nu+\mu+k+1\right)}{\Gamma% \left(\nu+k+\frac{3}{2}\right)}\frac{{\left(\mu+\frac{1}{2}\right)_{k}}}{k!}\*% \sin\left((\nu+\mu+2k+1)\theta\right),$

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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 ), $\sin\NVar{z}$ : 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 $\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\pi^{1/2}2^{\mu}(\sin\theta)^{% \mu}\*\sum_{k=0}^{\infty}\frac{\Gamma\left(\nu+\mu+k+1\right)}{\Gamma\left(\nu% +k+\frac{3}{2}\right)}\frac{{\left(\mu+\frac{1}{2}\right)_{k}}}{k!}\*\cos\left% ((\nu+\mu+2k+1)\theta\right).$

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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 ), $\sin\NVar{z}$ : 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 $\mathsf{P}_{n}\left(\cos\theta\right)=\frac{2^{2n+2}(n!)^{2}}{\pi(2n+1)!}\*% \sum_{k=0}^{\infty}\frac{1\cdot 3\cdots(2k-1)}{k!}\*\frac{(n+1)(n+2)\cdots(n+k% )}{(2n+3)(2n+5)\cdots(2n+2k+1)}\*\sin\left((n+2k+1)\theta\right),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, : factorial (as in ), $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\sin\NVar{z}$ : sine function, : 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 $\mathsf{Q}_{n}\left(\cos\theta\right)=\frac{2^{2n+1}(n!)^{2}}{(2n+1)!}\*\sum_{% k=0}^{\infty}\frac{1\cdot 3\cdots(2k-1)}{k!}\*\frac{(n+1)(n+2)\cdots(n+k)}{(2n% +3)(2n+5)\cdots(2n+2k+1)}\*\cos\left((n+2k+1)\theta\right),$

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Symbols:: $\cos\NVar{z}$ : cosine function, : factorial (as in ), $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, : 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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