14 Legendre and Related Functions Complex Arguments 14.27 Zeros 14.29 Generalizations

§14.28 Sums

ⓘ

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

ⓘ

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

ⓘ

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

ⓘ

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

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§14.28 Sums

Contents

§14.28(i) Addition Theorem

§14.28(ii) Heine’s Formula

§14.28(iii) Other Sums