14 Legendre and Related Functions Real Arguments 14.17 Integrals 14.19 Toroidal (or Ring) Functions

§14.18 Sums

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Keywords:: Ferrers functions, associated Legendre functions, sums
Permalink:: http://dlmf.nist.gov/14.18
See also:: Annotations for Ch.14

§14.18(i) Expansion Theorem
§14.18(ii) Addition Theorems
§14.18(iii) Other Sums
§14.18(iv) Compendia

§14.18(i) Expansion Theorem

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Keywords:: associated Legendre functions, expansions in series of
Permalink:: http://dlmf.nist.gov/14.18.i
See also:: Annotations for §14.18 and Ch.14

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).

§14.18(ii) Addition Theorems

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Keywords:: Ferrers functions, addition theorems, associated Legendre functions
Notes:: See Erdélyi et al. (1953a, pp. 168–169) and Olver (1997b, p. 183). Errors in Erdélyi et al. (1953a) have been corrected. (14.18.2) may be derived from (14.7.16), (14.9.3), and (14.18.1).
Permalink:: http://dlmf.nist.gov/14.18.ii
See also:: Annotations for §14.18 and Ch.14

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		$\mathsf{P}_{\nu}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2% }\cos\phi\right)=\mathsf{P}_{\nu}\left(\cos\theta_{1}\right)\mathsf{P}_{\nu}% \left(\cos\theta_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\mathsf{P}^{-m}_{\nu}% \left(\cos\theta_{1}\right)\mathsf{P}^{m}_{\nu}\left(\cos\theta_{2}\right)\cos% \left(m\phi\right),$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\cos\NVar{z}$ : cosine function, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $\nu$ : general degree, $\theta_{1}$ , $\theta_{2}$ : real and $\phi$ : real Referenced by: §14.18(ii), §14.18(ii) Permalink: http://dlmf.nist.gov/14.18.E1 Encodings: TeX, pMML, png See also: Annotations for §14.18(ii), §14.18 and Ch.14

14.18.2		$\mathsf{P}_{n}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}% \cos\phi\right)=\sum_{m=-n}^{n}(-1)^{m}\mathsf{P}^{-m}_{n}\left(\cos\theta_{1}% \right)\mathsf{P}^{m}_{n}\left(\cos\theta_{2}\right)\cos\left(m\phi\right).$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\cos\NVar{z}$ : cosine function, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $n$ : nonnegative integer, $\theta_{1}$ , $\theta_{2}$ : real and $\phi$ : real Referenced by: §14.18(ii), §14.18(ii) Permalink: http://dlmf.nist.gov/14.18.E2 Encodings: TeX, pMML, png See also: Annotations for §14.18(ii), §14.18 and Ch.14

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		$\mathsf{Q}_{\nu}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2% }\cos\phi\right)=\mathsf{P}_{\nu}\left(\cos\theta_{1}\right)\mathsf{Q}_{\nu}% \left(\cos\theta_{2}\right)+2\sum_{m=1}^{\infty}(-1)^{m}\mathsf{P}^{-m}_{\nu}% \left(\cos\theta_{1}\right)\mathsf{Q}^{m}_{\nu}\left(\cos\theta_{2}\right)\cos% \left(m\phi\right).$
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the second kind, $\cos\NVar{z}$ : cosine function, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)$ : Ferrers function of the second kind, $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $\nu$ : general degree, $\theta_{1}$ , $\theta_{2}$ : real and $\phi$ : real Referenced by: §14.18(ii), Ch.14 Permalink: http://dlmf.nist.gov/14.18.E3 Encodings: TeX, pMML, png See also: Annotations for §14.18(ii), §14.18 and Ch.14

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		$P_{\nu}\left(\cosh\xi_{1}\cosh\xi_{2}-\sinh\xi_{1}\sinh\xi_{2}\cos\phi\right)=% P_{\nu}\left(\cosh\xi_{1}\right)P_{\nu}\left(\cosh\xi_{2}\right)+2\sum_{m=1}^{% \infty}(-1)^{m}P^{-m}_{\nu}\left(\cosh\xi_{1}\right)P^{m}_{\nu}\left(\cosh\xi_% {2}\right)\cos\left(m\phi\right),$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : Legendre function of the first kind, $m$ : nonnegative integer, $\nu$ : general degree, $\phi$ : real and $\xi_{1}$ , $\xi_{2}$ : positive Referenced by: §14.18(ii) Permalink: http://dlmf.nist.gov/14.18.E4 Encodings: TeX, pMML, png See also: Annotations for §14.18(ii), §14.18 and Ch.14

14.18.5		$Q_{\nu}\left(\cosh\xi_{1}\cosh\xi_{2}-\sinh\xi_{1}\sinh\xi_{2}\cos\phi\right)=% P_{\nu}\left(\cosh\xi_{1}\right)Q_{\nu}\left(\cosh\xi_{2}\right)+2\sum_{m=1}^{% \infty}(-1)^{m}P^{-m}_{\nu}\left(\cosh\xi_{1}\right)Q^{m}_{\nu}\left(\cosh\xi_% {2}\right)\cos\left(m\phi\right).$
ⓘ Symbols: $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the second kind, $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : 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, $m$ : nonnegative integer, $\nu$ : general degree, $\phi$ : real and $\xi_{1}$ , $\xi_{2}$ : positive Referenced by: §14.18(ii) Permalink: http://dlmf.nist.gov/14.18.E5 Encodings: TeX, pMML, png See also: Annotations for §14.18(ii), §14.18 and Ch.14

§14.18(iii) Other Sums

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Notes:: See Erdélyi et al. (1953a, pp. 162 and 167). (14.18.8) may be derived from (14.7.17) and (14.18.9).
Referenced by:: Erratum (V1.0.7) for Subsection 14.18(iii)
Permalink:: http://dlmf.nist.gov/14.18.iii
Addition (effective with 1.0.7):: The first two equations in this subsection have been identified as Christoffel’s Formulas, and the two sentences following 14.18.7 have been added.
Suggested 2012-07-05 by Jonathon Connor
See also:: Annotations for §14.18 and Ch.14

Christoffel’s Formulas

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See also:: Annotations for §14.18(iii), §14.18 and Ch.14

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

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See also:: Annotations for §14.18(iii), §14.18 and Ch.14

14.18.8		$\mathsf{P}_{\nu}\left(-x\right)=\frac{\sin\left(\nu\pi\right)}{\pi}\sum_{n=0}^% {\infty}\frac{2n+1}{(\nu-n)(\nu+n+1)}\mathsf{P}_{n}\left(x\right),$
$\nu\notin\mathbb{Z}$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{Z}$ : set of all integers, $\notin$ : not an element of, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\sin\NVar{z}$ : sine function, $n$ : nonnegative integer and $\nu$ : general degree Referenced by: §14.18(iii) Permalink: http://dlmf.nist.gov/14.18.E8 Encodings: TeX, pMML, png See also: Annotations for §14.18(iii), §14.18(iii), §14.18 and Ch.14

Dougall’s Expansion

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Keywords:: Dougall’s expansion, associated Legendre functions
See also:: Annotations for §14.18(iii), §14.18 and Ch.14

14.18.9		$\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{\sin\left(\nu\pi\right)}{\pi}\sum_% {n=0}^{\infty}(-1)^{n}\frac{2n+1}{(\nu-n)(\nu+n+1)}\mathsf{P}^{-\mu}_{n}\left(% x\right),$
$-1<x\leq 1$ , $\mu\geq 0$ , $\nu\notin\mathbb{Z}$ .
ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{Z}$ : set of all integers, $\notin$ : not an element of, $\sin\NVar{z}$ : sine function, $n$ : nonnegative integer, $\mu$ : general order and $\nu$ : general degree Referenced by: §14.18(iii) Permalink: http://dlmf.nist.gov/14.18.E9 Encodings: TeX, pMML, png See also: Annotations for §14.18(iii), §14.18(iii), §14.18 and Ch.14

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

§14.18(iv) Compendia

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Keywords:: Ferrers functions, associated Legendre functions, sums
Referenced by:: §14.28(iii)
Permalink:: http://dlmf.nist.gov/14.18.iv
See also:: Annotations for §14.18 and Ch.14

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).

14.17 Integrals 14.19 Toroidal (or Ring) Functions

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