Multivariable recurrences with constant coefficients (Q2883414)

Recurrent sequence with constant coefficients in \(t\) variables with the form NEWLINE\[NEWLINEu(n_1, \ldots, n_t) = \sum_{(j_1, \ldots, j_t) \in J} a(j_1, \ldots, j_t)u(n_1- j_1, \ldots, n_t - j_t)NEWLINE\]NEWLINE are discussed, where \(J\) is a finite subset of \((\mathbb Z_0)^t - (0, 0, \ldots, 0)\), \(\mathbb Z_0 = \{0, 1, 2,\ldots \}\), and \(a(j_1, \ldots, j_t)\) are constants. It is proved that the three conditions on the \(t\)-dimensional array \(u(n_1,\ldots, n_t)\), \(0 \leq n_i\) are equivalent:NEWLINENEWLINENEWLINE1) \(u(n_1, ..., n_t) = a_{1,j}u(n_1, \ldots n_j - 1, \ldots, n_t) + a_{r_j,j}u(n_1, \ldots, n_j - r_j, \ldots, n_t)\) for \(1 \leq j \leq t\) and \(n_j \geq r_j\).NEWLINENEWLINENEWLINE2) \(u(n_1, \ldots, n_t) = \sum _{\substack{ (i_1, \ldots, i_t) \\ 1 \leq i_j \leq e_j }} B_{1,i_1}(n_1) \ldots B_{t,i_t}(n_t) \lambda_{1,i_1}^{n_1} \ldots \lambda_{t,i_t}^{n_t}\), where \(B_{j,i_j}\) is a polynomial of degree \(< d_{j,i_j}\).NEWLINENEWLINENEWLINE3) \(\sum\limits_{n_1, \ldots,n_t \geq 0} u(n_1, \ldots, n_t) x^{n_1}\ldots x^{n_t} = \frac{P(x_1, \ldots, x_t)}{Q_1(x_1) \ldots Q_t(x_t)},\) where \(P\) is a polynomial whose degree in \(x_j\) is less than \(r_j\).