14 Legendre and Related Functions Complex Arguments 14.27 Zeros 14.29 Generalizations

§14.28 Sums

Keywords:: associated Legendre functions
Referenced by:: §14.2(ii)
Permalink:: http://dlmf.nist.gov/14.28

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

§14.28(i) Addition Theorem

Keywords:: associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.28.i

When $\realpart{z_{1}}>0$ , $\realpart{z_{2}}>0$ , $|\mathop{\mathrm{ph}\/}\nolimits\!\left(z_{1}-1\right)|<\pi$ , and $|\mathop{\mathrm{ph}\/}\nolimits\!\left(z_{2}-1\right)|<\pi$ ,

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, : complex variable, : nonnegative integer and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.28.E1
Encodings:: TeX, pMML, png

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

§14.28(ii) Heine’s Formula

Notes:: See Erdélyi et al. (1953a, p. 168) or Olver (1997b, p. 473).
Keywords:: Heine’s formula, associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.28.ii

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

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, $\in$ : element of, : complex variable, : nonnegative integer and $\mathcal{E}_{1}$ , $\mathcal{E}_{2}$ : ellipses
Permalink:: http://dlmf.nist.gov/14.28.E2
Encodings:: TeX, pMML, png

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 $\mathcal{E}_{2}$ .

§14.28(iii) Other Sums

Permalink:: http://dlmf.nist.gov/14.28.iii

See §14.18(iv).

§14.28 Sums

Contents

§14.28(i) Addition Theorem

§14.28(ii) Heine’s Formula

§14.28(iii) Other Sums