On the derivatives $\partial^{2}P_{\nu}(z)/\partial\nu^{2}$ and $\partial Q_{\nu}(z)/\partial\nu$ of the Legendre functions with respect to their degrees

TL;DR

This paper derives explicit closed-form formulas for the second derivative of Legendre functions of the first kind and the first derivative of the second kind with respect to their degree, involving polynomials, dilogarithms, and logarithms.

Contribution

It provides new explicit formulas for derivatives of Legendre functions with respect to degree, including polynomial representations and special functions, filling a gap in the literature.

Findings

Closed-form expressions for degree derivatives of Legendre functions.

Polynomial representations of auxiliary functions involved.

Explicit formulas involving dilogarithm and logarithmic functions.

Abstract

We provide closed-form expressions for the degree-derivatives $[\partial^{2}P_{\nu}(z)/\partial\nu^{2}]_{\nu=n}$ and $[\partial Q_{\nu}(z)/\partial\nu]_{\nu=n}$, with $z\in\mathbb{C}$ and $n\in\mathbb{N}_{0}$, where $P_{\nu}(z)$ and $Q_{\nu}(z)$ are the Legendre functions of the first and the second kind, respectively. For $[\partial^{2}P_{\nu}(z)/\partial\nu^{2}]_{\nu=n}$, we find that $$\displaystyle\frac{\partial^{2}P_{\nu}(z)}{\partial\nu^{2}}\bigg|_{\nu=n}=-2P_{n}(z)\textrm{Li}_{2}\frac{1-z}{2}+B_{n}(z)\ln\frac{z+1}{2}+C_{n}(z),$$ 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 $[\partial Q_{\nu}(z)/\partial\nu]_{\nu=n}$ and $z\in\mathbb{C}\setminus[-1,1]$, we derive $$\displaystyle \frac{\partial Q_{\nu}(z)}{\partial\nu}\bigg|_{\nu=n}=-P_{n}(z)\textrm{Li}_{2}\frac{1-z}{2}-\frac{1}{2}P_{n}(z)\ln\frac{z+1}{2}\ln\frac{z-1}{2} +\frac{1}{4}B_{n}(z)\ln\frac{z+1}{2}-\frac{(-1)^{n}}{4}B_{n}(-z)\ln\frac{z-1}{2}-\frac{\pi^{2}}{6}P_{n}(z) +\frac{1}{4}C_{n}(z)-\frac{(-1)^{n}}{4}C_{n}(-z).$$ A counterpart expression for $[\partial Q_{\nu}(x)/\partial\nu]_{\nu=n}$, 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.