§14.13 Trigonometric Expansions

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

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

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

with conditional convergence for each.

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