Properties of associated Legendre conical functions

In cosmological fluctuations around a space of constant negative 3-curvature the relevant basis modes are associated Legendre conical functions. To facilitate the evaluation of the cosmic microwave background temperature anisotropy in the negative 3-curvature case we present some new properties of associated Legendre conical functions of the first and second kind,In particular we show that with the τwhere n ranges from -ℓ to ℓ in unit steps when K is a non-negative integer ℓ, and from -∞ to ∞ in unit steps otherwise. Also we can set, where n ranges from 0 to 2ℓ in unit steps when K is a non-negative integer ℓ, and from 0 to ∞ in unit steps otherwise. With these forms isolating the entire τ dependence, and especially its associated pole structure, we can use these forms to determine closed form expressions for integrals over τ of associated Legendre conical functions and their products. The QWe show how to use the divergence of this integral outside of this range in order to characterize the complex τ plane pole structure of Q -1/2-K -1/2+iτ (χ). We present a new treatment of the Borwein integral. We present both a new derivation of and a new generalization of the Nyquist-Shannon sampling theorem that is used in signal processing. We apply this analysis to the cosmic microwave background temperature anisotropy, to thus uncover an unexpected connection between signal processing and cosmological perturbation theory.