# §14.28 Sums

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

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

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

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).