Two-geometric-coefficient expansion of a causal ARMA(2,1) process (${\psi }_j=a{r}^j+d{s}^j$)

When the transfer function is $(1+\theta z)/((1-rz)(1-sz))$ with distinct $r,s$ of modulus less than one, partial fractions give coefficients ${\psi }_j=a{r}^j+d{s}^j$, where $a+d=1$ and $-as-dr=\theta$. White-noise orthogonality gives covariance at lag $h\ge 0$ as ${\sigma }^2[a^2{r}^h/(1-r^2)+d^2{s}^h/(1-s^2)+ad(r^h+s^h)/(1-rs)]$. This combines stable recursion with a closed geometric-sum covariance.