The parameter derivatives $[\partial^{2}P_{\nu}(z)/\partial\nu^{2}]_{\nu=0}$ and $[\partial^{3}P_{\nu}(z)/\partial\nu^{3}]_{\nu=0}$, where $P_{\nu}(z)$ is the Legendre function of the first kind

Abstract

We derive explicit expressions for the parameter derivatives $[\partial^{2}P_{\nu}(z)/\partial\nu^{2}]_{\nu=0}$ and $[\partial^{3}P_{\nu}(z)/\partial\nu^{3}]_{\nu=0}$, where $P_{\nu}(z)$ is the Legendre function of the first kind. It is found that {displaymath} \frac{\partial^{2}P_{\nu}(z)}{\partial\nu^{2}}\bigg|_{\nu=0} =-2\Li_{2}\frac{1-z}{2}, {displaymath} where $\Li_{2}z$ is the dilogarithm (this formula has been recently arrived at by Schramkowski using \emph{Mathematica}), and that {displaymath} \frac{\partial^{3}P_{\nu}(z)}{\partial\nu^{3}}\bigg|_{\nu=0} =12\Li_{3}\frac{z+1}{2}-6\ln\frac{z+1}{2}\Li_{2}\frac{z+1}{2} -\pi^{2}\ln\frac{z+1}{2}-12\zeta(3), {displaymath} where $\Li_{3}z$ is the polylogarithm of order 3 and $\zeta(s)$ is the Riemann zeta function.