arXiv 1301.65862013-01-28math.CAmath-phmath.MP

参数导数$[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=0}$和$[\partial^{3}P_ν(z)/\partialν^{3}]_{ν=0}$，其中$P_ν(z)$是第一类勒让德函数

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

Radosław Szmytkowski

备注：: 5 pages

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摘要：

我们推导出明确的表达式的参数导数$[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=0}$和$[\partial^{3}P_ν(z)/\partialν^{3}]_{ν=0}$，其中$P_ν(z)$是第一类的勒让德函数。我们发现{displaymath} \frac{\partial^{2}P_ν(z)}{\partialν^{2}} \bigg|_{ν=0} =-2 \Li_{2} \frac{1-z}{2}，{displaystyle}其中$\Li_{2}z$是双对数（Schramkowski最近使用\emph{Mathematica}得出了此公式），{displaystyle} \frac{\partial^{3}P_ν(z)}{\partialν^{3}} \bigg|_{v =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} -12 <$（3），{displaystyle}其中$\Li_{3}z$是3阶的多重对数，$ζ(s)$是黎曼zeta函数。

英文摘要：

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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