We build a process in order to extend the truncated weighted shift, using techniques of the bi-indexed recursive sequences. We apply this process to solve the subnormality of $2$-variable weighted shifts, whose associated moment sequence is a bi-indexed recursive sequence. Notably, we detail the case of the truncated $2$-variable weighted shift $T\equiv(T_1, T_2)$ of order $(2,2)$.
"Extension of the truncated bi-indexed weighted shifts, recursiveness and subnormality." Adv. Oper. Theory 4 (4) 836 - 851, Autumn 2019. https://doi.org/10.15352/aot.1812-1442