= Innovation coefficients of an ARMA(1,q) process
{title2=$c_j=\sum_{k=0}^{\min(j,q)}\theta_k\phi^{j-k}$}
With $\theta_0=1$ and $|\phi|<1$, expand the transfer function $\Theta(z)/(1-\phi z)$ by its geometric series. Its coefficient of $z^j$ is the displayed expression; after $j=q$ the coefficients follow a geometric tail. Causality gives $\operatorname{Cov}(\epsilon_{t-i},X_{t-k})=\sigma^2c_{i-k}$ for $i\ge k$, and zero otherwise. Hence $\gamma_k-\phi\gamma_{k-1}=\sigma^2\sum_{i=k}^q\theta_i c_{i-k}$ for $0\le k\le q$, with $\gamma_{-1}=\gamma_1$ in the zero-lag equation.
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