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

Research paper by Radosław Szmytkowski

Indexed on: 08 Jul '17Published on: 08 Jul '17Published in: arXiv - Mathematics - Classical Analysis and ODEs

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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.