HADAMARD INVERTIBILITY OF LINEARLY RECURSIVE SEQUENCES IN SEVERAL VARIABLES

A linearly recursive sequence in n variables is a tableau of scalars (fi(l)...i(n)) for i1,i,...,i(n) greater than or equal to O, such that for each 1 less than or equal to i less than or equal to n, all rows parallel to the ith axis satisfy a fixed linearly recursive relation h(i)(x) with constant coefficients. We show that such a tableau is Hadamard invertible (i.e., the tableau (1/fi(1)...i(n)) is linearly recursive) if and only if all fi(1)...i(n) not equivalent to O, and each row is eventually an interlacing of geometric sequences. The procedure is effective, i.e., given a linearly recursive sequence f=(fi(1)...i(n)), it can be tested for Hadamard invertibility by a finite algorithm. These results extend the case n=1 of Larson and Taft.