# §14.13 Trigonometric Expansions

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

 14.13.1 $\displaystyle\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta\right)$ $\displaystyle=\frac{2^{\mu+1}(\mathop{\sin\/}\nolimits\theta)^{\mu}}{\pi^{1/2}% }\*\sum_{k=0}^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+k+1% \right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu+k+\frac{3}{2}\right)}\frac{% \left(\mu+\frac{1}{2}\right)_{k}}{k!}\*\mathop{\sin\/}\nolimits\!\left((\nu+% \mu+2k+1)\theta\right),$ 14.13.2 $\displaystyle\mathop{\mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\theta\right)$ $\displaystyle=\pi^{1/2}2^{\mu}(\mathop{\sin\/}\nolimits\theta)^{\mu}\*\sum_{k=% 0}^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+k+1\right)}{\mathop% {\Gamma\/}\nolimits\!\left(\nu+k+\frac{3}{2}\right)}\frac{\left(\mu+\frac{1}{2% }\right)_{k}}{k!}\*\mathop{\cos\/}\nolimits\!\left((\nu+\mu+2k+1)\theta\right).$

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

with conditional convergence for each.

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