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

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 $$\frac{{\partial }^2{P}_{\nu }(z)}{\partial {\nu }^2}{|}_{\nu =n}=-2P_n(z){\text{Li}}_2\frac{1-z}{2}+B_n(z)\ln \frac{z+1}{2}+C_n(z),$$ where ${\text{Li}}_2[(1-z)/2]$ 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 $$\frac{\partial Q_{\nu }(z)}{\partial \nu }{|}_{\nu =n}=-P_n(z){\text{Li}}_2\frac{1-z}{2}-\frac{1}{2}{P}_n(z)\ln \frac{z+1}{2}\ln \frac{z-1}{2}+\frac{1}{4}{B}_n(z)\ln \frac{z+1}{2}-\frac{(-1{)}^n}{4}{B}_n(-z)\ln \frac{z-1}{2}-\frac{{\pi }^2}{6}{P}_n(z)+\frac{1}{4}{C}_n(z)-\frac{(-1{)}^n}{4}{C}_n(-z).$$ 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.

15 pages

References in corpus (1)

• Derivatives with respect to the order of the Legendre Polynomials