Yajun Zhou
Published 2013-04-05, updated 2013-05-24Version 2
The definite integrals $ \int_{-1}^1(1-x^2)^{(\nu-1)/2}[P_\nu(x)]^3\D x$, $ \int_{-1}^1(1-x^2)^{(\nu-1)/2}[P_\nu(x)]^2P_{\nu}(-x)\D x$, $\int_{-1}^1x(1-x^2)^{(\nu-1)/2}[P_{\nu+1}(x)]^3\D x $ and $\int_{-1}^1x(1-x^2)^{(\nu-1)/2}[P_{\nu+1}(x)]^2P_{\nu+1}(-x)\D x $ are evaluated in closed form, where $P_\nu$ is the Legendre function of degree $\nu$, and $ \R\nu>-1$. Special cases of these formulae are related to certain integrals over elliptic integrals that have arithmetic interest.
Comments: Revised according to referee's report, 9 pages
Two Definite Integrals Involving Products of Four Legendre Functions
The parameter derivatives $[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=0}$ and $[\partial^{3}P_ν(z)/\partialν^{3}]_{ν=0}$, where $P_ν(z)$ is the Legendre function of the first kind
On the curious series related to the elliptic integrals
![QR code for arXiv:1304.1606 [math.CA]](http://oss.arxitics.com/images/favicon.svg)