Properties of associated Legendre conical functions

Abstract

We present some new properties of associated Legendre conical functions of the first and second kind, $P_{-1/2+i\ddot{I}„}^{-1/2-K}(\ddot{I}‡)$ and $Q_{-1/2+i\ddot{I}„}^{-1/2-K}(\ddot{I}‡)$. In particular we show that with the $\ddot{I}„$-independent ${\mathcal{R}}_n^K(\ddot{I}‡)=(2\ddot{I}€{)}^{-3/2}{\tanh }^{-K}\ddot{I}‡{\sinh }^{-1/2}\ddot{I}‡[\hat{I}“(1+K){]}^{-1}{\int }_0^{2\ddot{I}€}d\ddot{I}‰{\left(1-\cos \ddot{I}‰/\cosh \ddot{I}‡\right)}^K{e}^{in\ddot{I}‰}$ for any general $K$, we can set $P_{-1/2+i\ddot{I}„}^{-1/2-K}(\ddot{I}‡)=2{\sum }_n{\mathcal{R}}_n^K(\ddot{I}‡)\sin [(\ddot{I}„-in)\ddot{I}‡)]/(\ddot{I}„-in)$, where $n$ ranges from $-\ell$ to $\ell$ in unit steps when $K$ is a non-negative integer $\ell$, and from $-\infty$ to $\infty$ in unit steps otherwise. Also we can set $Q_{-1/2+i\ddot{I}„}^{-1/2-K}(\ddot{I}‡)=-\ddot{I}€{\sum }_n{\mathcal{R}}_{n-K}^K(\ddot{I}‡)e^{-i\ddot{I}‡(\ddot{I}„-i(n-K))}/(\ddot{I}„-i(n-K))$, where $n$ ranges from $0$ to $2\ell$ in unit steps when $K$ is a non-negative integer $\ell$, and from $0$ to $\infty$ in unit steps otherwise. With these forms isolating the entire $\ddot{I}„$ dependence, and especially its associated pole structure, we can use these forms to determine closed form expressions for integrals over $\ddot{I}„$ of associated Legendre conical functions and their products. The $Q_{-1/2+i\ddot{I}„}^{-1/2-K}(\ddot{I}‡)$ have an integral representation containing the integral ${\int }_{\ddot{I}}{‡}^{\infty }d\ddot{I}‰e^{i\ddot{I}‰\ddot{I}„}(\cosh \ddot{I}‰-\cosh \ddot{I}‡{)}^K$, an integral that only converges at $\ddot{I}‰=\infty$ if $\mathrm{Re}[K]<\mathrm{Im}[\ddot{I}„]$. We show how to use the divergence of this integral outside of this range in order to characterize the complex $\ddot{I}„$ plane pole structure of $Q_{-1/2+i\ddot{I}„}^{-1/2-K}(\ddot{I}‡)$. We present a new treatment of the Borwein integral and the Nyquist-Shannon sampling theorem.

22 pages