= Solution
Put $\theta_0=1$ and define the <autoregressive polynomial> and <moving-average polynomial> by
$$
\Phi(z)=1-\sum_{k=1}^p\phi_kz^k,\qquad\Theta(z)=\sum_{k=0}^q\theta_kz^k.
$$
For a minimal, coprime <ARMA> representation, the usual simultaneous <causality and invertibility root criteria for an ARMA model> are
$$
\boxed{\Phi(z)\ne0\text{ and }\Theta(z)\ne0\quad\text{for all }|z|\le1.}
$$
The autoregressive condition makes $\Theta/\Phi$ a stable one-sided filter and gives the causal second-order <stationary process>. The moving-average condition makes its inverse a stable one-sided filter, so the <white noise> can be reconstructed from present and past observations. If the two polynomials have a common factor, first use <common factor cancellation in an ARMA model>; necessary-and-sufficient root statements apply to the reduced noise-driven representation. Stationarity here is second-order stationarity; <weak white noise> alone does not assert strict stationarity of all finite-dimensional laws.
For $p=1$, write $\phi=\phi_1$, so $|\phi|<1$. Expanding $(1-\phi z)^{-1}=\sum_{r\ge0}\phi^rz^r$ gives the <Wold representation>
$$
X_t=\sum_{j\ge0}c_j\epsilon_{t-j},\qquad\boxed{c_j=\sum_{k=0}^{\min(j,q)}\theta_k\phi^{j-k}.}
$$
In particular $c_0=1$, $c_j=\phi c_{j-1}+\theta_j$ for $1\le j\le q$, and $c_j=\phi^{j-q}c_q$ for $j\ge q$. The coefficients are square-summable, so this series converges in mean square. Causality and invertibility identify the driver with the <linear innovation process>, justifying the Wold terminology. For $\phi=0$, powers with exponent zero equal one and the formula reduces to the finite moving average.
To obtain the <covariance> equations, subtract $\phi X_{t-1}$ and take <covariance> with $X_{t-k}$. Orthogonality of the <white noise> and the causal expansion imply
$$
\operatorname{Cov}(\epsilon_{t-i},X_{t-k})=\begin{cases}\sigma^2c_{i-k},&i\ge k,\\0,&i<k.\end{cases}
$$
Thus the <innovation coefficients of an ARMA(1,q) process> give the entire triangular system in one formula:
$$
\boxed{\gamma_k-\phi\gamma_{k-1}=\sigma^2\sum_{i=k}^q\theta_i c_{i-k}\quad(0\le k\le q),\qquad\gamma_k-\phi\gamma_{k-1}=0\quad(k>q).}
$$
At $k=0$, use $\gamma_{-1}=\gamma_1$, since the <autocovariance> of a real <stationary process> is symmetric. The terms are consequently $\sigma^2(c_0+\theta_1c_1+\cdots+\theta_qc_q)$ at zero, $\sigma^2\theta_qc_0$ at $q$, and the intervening partial sums as required.
For $q=1$, put $\theta=\theta_1$ and $c_1=\phi+\theta$. The equations become
$$
\gamma_0-\phi\gamma_1=\sigma^2(1+\theta\phi+\theta^2),\quad\gamma_1-\phi\gamma_0=\sigma^2\theta,\quad\gamma_k=\phi\gamma_{k-1}\ (k\ge2).
$$
Substitution of the second into the first, followed by the geometric recursion, yields the <autocovariance of a causal ARMA(1,1) process>:
$$
\boxed{\gamma_0=\frac{\sigma^2(1+\theta^2+2\phi\theta)}{1-\phi^2},\qquad\gamma_k=\frac{\sigma^2(\phi+\theta)(1+\phi\theta)}{1-\phi^2}\phi^{|k|-1}\quad(k\ne0).}
$$
This includes $\phi=0$: only lags $\pm1$ survive, as for an MA(1) process.
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