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

TL;DR

This paper derives explicit formulas for the second and third derivatives of Legendre functions of the first kind with respect to their order at zero, expressing them in terms of polylogarithms and zeta functions.

Contribution

It provides new explicit expressions for parameter derivatives of Legendre functions at zero, linking them to polylogarithms and special constants, extending previous results.

Findings

Explicit formulas for second and third derivatives at zero

Expressions involve dilogarithm, trilogarithm, and zeta functions

Results confirm recent numerical findings by Schramkowski

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.