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

§14.13 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),
$\pm\frac{1}{2}\pi i\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(% \mathop{\cos\/}\nolimits\theta\right)+\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}% \nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)=\pi^{\frac{1}{2}}% \mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)(2\mathop{\sin\/}\nolimits% \theta)^{\mu}e^{{\pm(\nu+\mu+1)i\theta}}\*\mathop{\mathbf{F}\/}\nolimits\!% \left(\nu+\mu+1,\mu+\frac{1}{2};\nu+\frac{3}{2};e^{{\pm 2i\theta}}\right).$
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.
Keywords:: Ferrers functions
Referenced by:: Other Changes
Permalink:: http://dlmf.nist.gov/14.13
Clarification (effective with 1.0.1):: The conditions for convergence of the expansions in this subsection were clarified.
Reported 2011-03-02

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

14.13.1 $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits% \theta\right)=\frac{2^{{\mu+1}}(\mathop{\sin\/}\nolimits\theta)^{{\mu}}}{\pi^{% {1/2}}}\*\sum_{{k=0}}^{{\infty}}\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+% \mu+k+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu+k+\frac{3}{2}\right)}% \frac{\left(\mu+\frac{1}{2}\right)_{{k}}}{k!}\*\mathop{\sin\/}\nolimits\!\left% ((\nu+\mu+2k+1)\theta\right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\cos\/}\nolimits z$ : cosine function, : : factorial, $\mathop{\sin\/}\nolimits z$ : sine function, $\mu$ : general order, $\nu$ : general degree and $0<\theta<\pi$ : variable
A&S Ref:: 8.7.1
Referenced by:: §14.13
Permalink:: http://dlmf.nist.gov/14.13.E1
Encodings:: TeX, pMML, png

14.13.2 $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits% \theta\right)=\pi^{{1/2}}2^{{\mu}}(\mathop{\sin\/}\nolimits\theta)^{{\mu}}\*% \sum_{{k=0}}^{{\infty}}\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+k+1% \right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu+k+\frac{3}{2}\right)}\frac{% \left(\mu+\frac{1}{2}\right)_{{k}}}{k!}\*\mathop{\cos\/}\nolimits\!\left((\nu+% \mu+2k+1)\theta\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\cos\/}\nolimits z$ : cosine function, : : factorial, $\mathop{\sin\/}\nolimits z$ : sine function, $\mu$ : general order, $\nu$ : general degree and $0<\theta<\pi$ : variable
A&S Ref:: 8.7.2
Referenced by:: §14.13
Permalink:: http://dlmf.nist.gov/14.13.E2
Encodings:: TeX, pMML, png

These Fourier series converge absolutely when $\realpart{\mu}<0$ , and conditionally when $\nu$ is real and $0\leq\mu<\frac{1}{2}$ .

In particular,

14.13.3 $\mathop{\mathsf{P}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta% \right)=\frac{2^{{2n+2}}(n!)^{2}}{\pi(2n+1)!}\*\sum_{{k=0}}^{{\infty}}\frac{1% \cdot 3\cdots(2k-1)}{k!}\*\frac{(n+1)(n+2)\cdots(n+k)}{(2n+3)(2n+5)\cdots(2n+2% k+1)}\*\mathop{\sin\/}\nolimits\!\left((n+2k+1)\theta\right),$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, : : factorial, $\mathop{\sin\/}\nolimits z$ : 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

14.13.4 $\mathop{\mathsf{Q}_{{n}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta% \right)=\frac{2^{{2n+1}}(n!)^{2}}{(2n+1)!}\*\sum_{{k=0}}^{{\infty}}\frac{1% \cdot 3\cdots(2k-1)}{k!}\*\frac{(n+1)(n+2)\cdots(n+k)}{(2n+3)(2n+5)\cdots(2n+2% k+1)}\*\mathop{\cos\/}\nolimits\!\left((n+2k+1)\theta\right),$

Symbols:: $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, : : factorial, : 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

with conditional convergence for each.

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

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