Innovation coefficients of an ARMA(1,q) process ($c_j={\sum }_{k=0}^{\min (j,q)}{\theta }_k{\varphi }^{j-k}$)

With ${\theta }_0=1$ and $|\varphi |<1$, expand the transfer function $\Theta (z)/(1-\varphi z)$ by its geometric series. Its coefficient of $z^j$ is the displayed expression; after $j=q$ the coefficients follow a geometric tail. Causality gives $\mathrm{Cov}({ϵ}_{t-i},X_{t-k})={\sigma }^2{c}_{i-k}$ for $i\ge k$, and zero otherwise. Hence ${\gamma }_k-\varphi {\gamma }_{k-1}={\sigma }^2{\sum }_{i=k}^q{\theta }_i{c}_{i-k}$ for $0\le k\le q$, with ${\gamma }_{-1}={\gamma }_1$ in the zero-lag equation.