Gegenbauer processes and their generalizations represent a general way of modeling long memory and seasonal long memory; they include ARFIMA, seasonal ARFIMA, and GARMA processes as special cases. Models from this class of processes have been used extensively in economics, finance, and in the physical sciences. An obstacle to using this class of models is in finding explicit formulas for the autocovariances that are valid for all lags. We provide a computationally efficient method of computing these autocovariances, by determining the moving average representation of these processes, and also give an asymptotic formula for the determinant of the covariance matrix. This allows feasible calculation of the exact Gaussian likelihood, while also making simulation, forecasting, and signal extraction practicable. The techniques are illustrated using maximum likelihood estimation to model atmospheric $\mbox{CO}_2$ data.