Properties of associated Legendre conical functions

TL;DR

This paper explores new mathematical properties of associated Legendre conical functions, providing integral representations and pole structure analysis, with applications to integral evaluations and classical sampling theorems.

Contribution

It introduces novel integral forms and pole analysis for associated Legendre conical functions, enhancing understanding of their complex structure and applications.

Findings

Derived new integral representations for Legendre conical functions.

Analyzed the pole structure of these functions in the complex plane.

Connected the functions' properties to classical sampling theorems.

Abstract

We present some new properties of associated Legendre conical functions of the first and second kind, $P^{-1/2-K}_{-1/2+i \tau}(\chi)$ and $Q^{-1/2-K}_{-1/2+i \tau}(\chi)$. In particular we show that with the $\tau$-independent $\mathcal{R}^{K}_{n}(\chi)=(2\pi)^{-3/2}\tanh^{-K}\chi\sinh^{-1/2}\chi[\Gamma(1+K)]^{-1}\int_{0}^{2\pi} d\omega \left (1-\cos \omega/\cosh\chi\right)^{K}e^{in\omega}$ for any general $K$, we can set $P^{-1/2-K}_{-1/2+i \tau}(\chi)=2\sum_{n}\mathcal{R}^{K}_{n}(\chi)\sin[(\tau-in)\chi)]/(\tau-in)$, 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 $Q^{-1/2-K}_{-1/2+i \tau}(\chi)=-\pi\sum_n\mathcal{R}^K_{n-K}(\chi)e^{-i \chi(\tau-i(n-K))}/(\tau-i(n-K))$, 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 $\tau$ dependence, and especially its associated pole structure, we can use these forms to determine closed form expressions for integrals over $\tau$ of associated Legendre conical functions and their products. The $Q^{-1/2-K}_{-1/2+i \tau}(\chi)$ have an integral representation containing the integral $\int_{\chi}^{\infty}d\omega e^{i\omega\tau}(\cosh\omega-\cosh\chi)^{K}$, an integral that only converges at $\omega=\infty$ if ${\rm Re}[K]<{\rm Im}[\tau]$. We show how to use the divergence of this integral outside of this range in order to characterize the complex $\tau$ plane pole structure of $Q^{-1/2-K}_{-1/2+i \tau}(\chi)$. We present a new treatment of the Borwein integral and the Nyquist-Shannon sampling theorem.