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

We derive explicit expressions for 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. It is found that {displaymath} \frac{\partial^{2}P_ν(z)}{\partialν^{2}}\bigg|_{ν=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_ν(z)}{\partialν^{3}}\bigg|_{ν=0} =12\Li_{3}\frac{z+1}{2}-6\ln\frac{z+1}{2}\Li_{2}\frac{z+1}{2} -π^{2}\ln\frac{z+1}{2}-12ζ(3), {displaymath} where $\Li_{3}z$ is the polylogarithm of order 3 and $ζ(s)$ is the Riemann zeta function.

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