Some integrals and series involving the Gegenbauer polynomials and the Legendre functions on the cut (-1,1)

We use the recent findings of Cohl [arXiv:1105.2735] and evaluate two integrals involving the Gegenbauer polynomials: $\int_{-1}^{x}\mathrm{d}t\:(1-t^{2})^{λ-1/2}(x-t)^{-κ-1/2}C_{n}^λ(t)$ and $\int_{x}^{1}\mathrm{d}t\:(1-t^{2})^{λ-1/2}(t-x)^{-κ-1/2}C_{n}^λ(t)$, both with $\Realλ>-1/2$, $\Realκ<1/2$, $-1<x<1$. The results are expressed in terms of the on-the-cut associated Legendre functions $P_{n+λ-1/2}^{κ-λ}(\pm x)$ and $Q_{n+λ-1/2}^{κ-λ}(x)$. In addition, we find closed-form representations of the series $\sum_{n=0}^{\infty}(\pm)^{n}[(n+λ)/λ]P_{n+λ-1/2}^{κ-λ}(\pm x)C_{n}^λ(t)$ and $\sum_{n=0}^{\infty}(\pm)^{n}[(n+λ)/λ]Q_{n+λ-1/2}^{κ-λ}(\pm x)C_{n}^λ(t)$, both with $\Realλ>-1/2$, $\Realκ<1/2$, $-1<t<1$, $-1<x<1$.

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