Poisson-kernel expansion of an AR(1) spectrum (${\sum }_{k\in \mathbb{Z}}{\varphi }^{|k|}{z}^k=\frac{1-{\varphi }^2}{(1-\varphi z)(1-\varphi z^{-1})}$)

For $|\varphi |<1$, the bilateral sum ${\sum }_{k\in \mathbb{Z}}{\varphi }^{|k|}{z}^k$ equals $(1-{\varphi }^2)/((1-\varphi z)(1-\varphi z^{-1}))$ on the unit circle. Multiplying by ${\sigma }^2/(\pi (1-{\varphi }^2))$ gives the one-sided autoregressive spectrum and reads off its covariance coefficients.