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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 % \begin{displaymath} \frac{\partial^{2}P_{\nu}(z)}{\partial\nu^{2}}\bigg|_{\nu=n} =-2P_{n}(z)\Li_{2}\frac{1-z}{2}+B_{n}(z)\ln\frac{z+1}{2}+C_{n}(z), \end{displaymath} % where $\Li_{2}[(1-z)/2]$ 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 % \begin{eqnarray*} \frac{\partial Q_{\nu}(z)}{\partial\nu}\bigg|_{\nu=n} &=& -P_{n}(z)\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} \nonumber \\ && -\,\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). \end{eqnarray*} % 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.
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- Category:
- Articles
- Type:
- artykuł w czasopiśmie wyróżnionym w JCR
- Published in:
-
INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS
no. 28,
edition 9,
pages 645 - 662,
ISSN: 1065-2469 - Language:
- English
- Publication year:
- 2017
- Bibliographic description:
- Szmytkowski R.: 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// INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS. -Vol. 28, iss. 9 (2017), s.645-662
- DOI:
- Digital Object Identifier (open in new tab) 10.1080/10652469.2017.1339039
- Verified by:
- Gdańsk University of Technology
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