A representation of recurrent sequences by previous terms (Q2883399)

A recurrent sequence of order \(m\) can be defined as a sequence with arbitrary real or complex numbers \(u_0, u_1, \dots, u_m, \dots\), and for \(k \geq m\): NEWLINE\[NEWLINEu_k = C_1u_{k-1} + C_2u_{k-2} + \dots + C_mu_{k-m},NEWLINE\]NEWLINE where \(C_1, C_2, \dots , C_m\) are arbitrary real numbers and \(C_m \not= 0\).NEWLINENEWLINELet \(a_1, a_2, \dots, a_m\) be arbitrary natural numbers such that \(1 \leq a_1 < a_2 < \dots < a_m\). In the paper, an algorithm for finding the numbers \(b_1, b_2, \dots, b_m\) is given, such that NEWLINE\[NEWLINEu_{k+a_m} = b_1u_{k+a_{m-1}} + b_2u_{k+a_{m-2}} + \dots + b_{m-1}u_{k+a_{1}} + b_mu_k.NEWLINE\]