A closed formula for the durbin-levinsonʹs algorithm in seasonal fractionally integrated processes

We consider the fractionally integrated ARFIMA Processes with seasonality s, denoted by SARFIMA (0, D, 0)s. This work presents a closed formula for the Durbin-Levinsonʹs algorithm relating the partial autocorrelation and the autocorrelation functions of these processes. In order to obtain the closed formula we show a hypergeometric identity, namely ( l − D ) ∑ j = 0 l − 1 ( j l − 1 ) Γ ( j − D ) Γ ( l − j − D ) Γ ( l − j + D ) Γ ( l − j − D + 1 ) = D Γ ( − D ) Γ ( D − l + 1 ) ( l − 1 ) ! 2 l − 1 ⋅ ∏ i = 0 l − 2 ( D − i + 1 2 ) , y nonnegative integer l and for any D ∈ (−0.5, 0.5). cursive algorithm that requires the use of the left-hand side of the above expression will have smaller error under the use of the right-hand side formula. rbin-Levinsonʹs algorithm is fully calculated for the SARFIMA (0, D, 0)s processes.