On the derivatives $\partial^{2}P_ν(z)/\partialν^{2}$ and $\partial Q_ν(z)/\partialν$ of the Legendre functions with respect to their degrees

Abstract

We provide closed-form expressions for the degree-derivatives $[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=n}$ and $[\partial Q_ν(z)/\partialν]_{ν=n}$, with $z\in\mathbb{C}$ and $n\in\mathbb{N}_{0}$, where $P_ν(z)$ and $Q_ν(z)$ are the Legendre functions of the first and the second kind, respectively. For $[\partial^{2}P_ν(z)/\partialν^{2}]_{ν=n}$, we find that $$\displaystyle\frac{\partial^{2}P_ν(z)}{\partialν^{2}}\bigg|_{ν=n}=-2P_{n}(z)\textrm{Li}_{2}\frac{1-z}{2}+B_{n}(z)\ln\frac{z+1}{2}+C_{n}(z),$$ where $\textrm{Li}_{2}[(1-z)/2]$ is the dilogarithm function, $P_{n}(z)$ is the Legendre polynomial, while $B_{n}(z)$ and $C_{n}(z)$ are certain polynomials in $z$ of degree $n$. For $[\partial Q_ν(z)/\partialν]_{ν=n}$ and $z\in\mathbb{C}\setminus[-1,1]$, we derive $$\displaystyle \frac{\partial Q_ν(z)}{\partialν}\bigg|_{ν=n}=-P_{n}(z)\textrm{Li}_{2}\frac{1-z}{2}-\frac{1}{2}P_{n}(z)\ln\frac{z+1}{2}\ln\frac{z-1}{2} +\frac{1}{4}B_{n}(z)\ln\frac{z+1}{2}-\frac{(-1)^{n}}{4}B_{n}(-z)\ln\frac{z-1}{2}-\frac{π^{2}}{6}P_{n}(z) +\frac{1}{4}C_{n}(z)-\frac{(-1)^{n}}{4}C_{n}(-z).$$ A counterpart expression for $[\partial Q_ν(x)/\partialν]_{ν=n}$, applicable when $x\in(-1,1)$, is also presented. Explicit representations of the polynomials $B_{n}(z)$ and $C_{n}(z)$ as linear combinations of the Legendre polynomials are given.