14 Legendre and Related Functions Complex Arguments 14.27 Zeros 14.29 Generalizations

§14.28 Sums

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Keywords:: associated Legendre functions, sums
Referenced by:: §14.2(ii)
Permalink:: http://dlmf.nist.gov/14.28
See also:: Annotations for Ch.14

§14.28(i) Addition Theorem
§14.28(ii) Heine’s Formula
§14.28(iii) Other Sums

§14.28(i) Addition Theorem

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Keywords:: addition theorems, associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.28.i
See also:: Annotations for §14.28 and Ch.14

When $\Re z_{1}>0$ , $\Re z_{2}>0$ , $|\operatorname{ph}\left(z_{1}-1\right)|<\pi$ , and $|\operatorname{ph}\left(z_{2}-1\right)|<\pi$ ,

14.28.1 $P_{\nu}\left(z_{1}z_{2}-\left(z_{1}^{2}-1\right)^{1/2}\left(z_{2}^{2}-1\right)% ^{1/2}\cos\phi\right)=P_{\nu}\left(z_{1}\right)P_{\nu}\left(z_{2}\right)+2\sum% _{m=1}^{\infty}(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m+1% \right)}\*P^{m}_{\nu}\left(z_{1}\right)P^{m}_{\nu}(z_{2})\cos\left(m\phi\right),$

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind, $\cos\NVar{z}$ : cosine function, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : Legendre function of the first kind, : complex variable, : nonnegative integer and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.28.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §14.28(i), §14.28 and Ch.14

where the branches of the square roots have their principal values when $z_{1},z_{2}\in(1,\infty)$ and are continuous when $z_{1},z_{2}\in\mathbb{C}\setminus(0,1]$ . For this and similar results see Erdélyi et al. (1953a, §3.11).

§14.28(ii) Heine’s Formula

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Keywords:: Heine’s formula, associated Legendre functions
Notes:: See Erdélyi et al. (1953a, p. 168) or Olver (1997b, p. 473).
Referenced by:: Erratum (V1.0.10) for References, Erratum (V1.0.9) for References
Permalink:: http://dlmf.nist.gov/14.28.ii
Addition (effective with 1.0.10):: An additional reference to Cohl (2013a) has been introduced into the final sentence of this subsection.
Addition (effective with 1.0.9):: A sentence and reference to Cohl (2013b) have been added at the end of this section.
Suggested 2014-05-22 by Howard Cohl
See also:: Annotations for §14.28 and Ch.14

14.28.2 $\sum_{n=0}^{\infty}(2n+1)Q_{n}\left(z_{1}\right)P_{n}\left(z_{2}\right)=\frac{% 1}{z_{1}-z_{2}},$ $z_{1}\in\mathcal{E}_{1}$ , $z_{2}\in\mathcal{E}_{2}$ ,

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Symbols:: $\in$ : element of, $P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$ : 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, : complex variable, : nonnegative integer and $\mathcal{E}_{1}$ , $\mathcal{E}_{2}$ : ellipses
Permalink:: http://dlmf.nist.gov/14.28.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §14.28(ii), §14.28 and Ch.14

where $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ are ellipses with foci at $\pm 1$ , $\mathcal{E}_{2}$ being properly interior to $\mathcal{E}_{1}$ . The series converges uniformly for $z_{1}$ outside or on $\mathcal{E}_{1}$ , and $z_{2}$ within or on $\mathcal{E}_{2}$ .

For generalizations in terms of Gegenbauer and Jacobi polynomials, see Theorem 2.1 in Cohl (2013b) and Theorem 1 in Cohl (2013a) respectively.

§14.28(iii) Other Sums

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Permalink:: http://dlmf.nist.gov/14.28.iii
See also:: Annotations for §14.28 and Ch.14

See §14.18(iv).

14.27 Zeros 14.29 Generalizations

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§14.28 Sums

Contents

§14.28(i) Addition Theorem

§14.28(ii) Heine’s Formula

§14.28(iii) Other Sums