§14.18 Sums

§14.18(i) Expansion Theorem

For expansions of arbitrary functions in series of Legendre polynomials see §18.18(i), and for expansions of arbitrary functions in series of associated Legendre functions see Schäfke (1961b).

In (14.18.1) and (14.18.2), $\theta_{1}$, $\theta_{2}$, and $\theta_{1}+\theta_{2}$ all lie in $[0,\pi)$, and $\phi$ is real.

 14.18.1 $\mathsf{P}_{\nu}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2% }\cos\phi\right)=\mathsf{P}_{\nu}\left(\cos\theta_{1}\right)\mathsf{P}_{\nu}% \left(\cos\theta_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\mathsf{P}^{-m}_{\nu}% \left(\cos\theta_{1}\right)\mathsf{P}^{m}_{\nu}\left(\cos\theta_{2}\right)\cos% \left(m\phi\right),$
 14.18.2 $\mathsf{P}_{n}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}% \cos\phi\right)=\sum_{m=-n}^{n}(-1)^{m}\mathsf{P}^{-m}_{n}\left(\cos\theta_{1}% \right)\mathsf{P}^{m}_{n}\left(\cos\theta_{2}\right)\cos\left(m\phi\right).$

In (14.18.3), $\theta_{1}$ lies in $(0,\frac{1}{2}\pi)$, $\theta_{2}$ and $\theta_{1}+\theta_{2}$ both lie in $(0,\pi)$, $\theta_{1}<\theta_{2}$, $\phi$ is real, and $\nu\neq-1,-2,-3,\dots$.

 14.18.3 $\mathsf{Q}_{\nu}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2% }\cos\phi\right)=\mathsf{P}_{\nu}\left(\cos\theta_{1}\right)\mathsf{Q}_{\nu}% \left(\cos\theta_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\mathsf{P}^{-m}_{\nu}% \left(\cos\theta_{1}\right)\mathsf{Q}^{m}_{\nu}\left(\cos\theta_{2}\right)\cos% \left(m\phi\right).$

In (14.18.4) and (14.18.5), $\xi_{1}$ and $\xi_{2}$ are positive, and $\phi$ is real; also in (14.18.5) $\xi_{1}<\xi_{2}$ and $\nu\neq-1,-2,-3,\dots$.

 14.18.4 $P_{\nu}\left(\cosh\xi_{1}\cosh\xi_{2}-\sinh\xi_{1}\sinh\xi_{2}\cos\phi\right)=% P_{\nu}\left(\cosh\xi_{1}\right)P_{\nu}\left(\cosh\xi_{2}\right)+2\sum_{m=1}^{% \infty}(-1)^{m}P^{-m}_{\nu}\left(\cosh\xi_{1}\right)P^{m}_{\nu}\left(\cosh\xi_% {2}\right)\cos\left(m\phi\right),$
 14.18.5 $Q_{\nu}\left(\cosh\xi_{1}\cosh\xi_{2}-\sinh\xi_{1}\sinh\xi_{2}\cos\phi\right)=% P_{\nu}\left(\cosh\xi_{1}\right)Q_{\nu}\left(\cosh\xi_{2}\right)+2\sum_{m=1}^{% \infty}(-1)^{m}P^{-m}_{\nu}\left(\cosh\xi_{1}\right)Q^{m}_{\nu}\left(\cosh\xi_% {2}\right)\cos\left(m\phi\right).$

§14.18(iii) Other Sums

Christoffel’s Formulas

 14.18.6 $\displaystyle(x-y)\sum_{k=0}^{n}(2k+1)P_{k}\left(x\right)P_{k}\left(y\right)$ $\displaystyle=(n+1)\left(P_{n+1}\left(x\right)P_{n}\left(y\right)-P_{n}\left(x% \right)P_{n+1}\left(y\right)\right),$ ⓘ Symbols: $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$: Legendre function of the first kind and $n$: nonnegative integer A&S Ref: 8.9.1 Referenced by: Erratum (V1.0.7) for Subsection 14.18(iii) Permalink: http://dlmf.nist.gov/14.18.E6 Encodings: TeX, pMML, png See also: Annotations for §14.18(iii), §14.18(iii), §14.18 and Ch.14 14.18.7 $\displaystyle(x-y)\sum_{k=0}^{n}(2k+1)P_{k}\left(x\right)Q_{k}\left(y\right)$ $\displaystyle=(n+1)\left(P_{n+1}\left(x\right)Q_{n}\left(y\right)-P_{n}\left(x% \right)Q_{n+1}\left(y\right)\right)-1.$ ⓘ Symbols: $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$: Legendre function of the first kind, $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$: Legendre function of the second kind and $n$: nonnegative integer A&S Ref: 8.9.2 Referenced by: §14.18(iii), Erratum (V1.0.7) for Subsection 14.18(iii) Permalink: http://dlmf.nist.gov/14.18.E7 Encodings: TeX, pMML, png See also: Annotations for §14.18(iii), §14.18(iii), §14.18 and Ch.14

In these formulas the Legendre functions are as in §14.3(ii) with $\mu=0$. The formulas are also valid with the Ferrers functions as in §14.3(i) with $\mu=0$.

Zonal Harmonic Series

 14.18.8 $\mathsf{P}_{\nu}\left(-x\right)=\frac{\sin\left(\nu\pi\right)}{\pi}\sum_{n=0}^% {\infty}\frac{2n+1}{(\nu-n)(\nu+n+1)}\mathsf{P}_{n}\left(x\right),$ $\nu\notin\mathbb{Z}$.

Dougall’s Expansion

 14.18.9 $\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{\sin\left(\nu\pi\right)}{\pi}\sum_% {n=0}^{\infty}(-1)^{n}\frac{2n+1}{(\nu-n)(\nu+n+1)}\mathsf{P}^{-\mu}_{n}\left(% x\right),$ $-1, $\mu\geq 0$, $\nu\notin\mathbb{Z}$.

For a series representation of the Dirac delta in terms of products of Legendre polynomials see (1.17.22).

§14.18(iv) Compendia

For collections of sums involving associated Legendre functions, see Hansen (1975, pp. 367–377, 457–460, and 475), Erdélyi et al. (1953a, §3.10), Gradshteyn and Ryzhik (2015, §8.92), Magnus et al. (1966, pp. 178–184), and Prudnikov et al. (1990, §§5.2, 6.5). See also §18.18 and (34.3.19).