# §14.28 Sums

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

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

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

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.

See §14.18(iv).