A new generalized Cassini determinant (Q2833640)

The hyperfibonacci sequences are defined by NEWLINE\[NEWLINEF_n^{(r)} = \sum_{k=0}^n F_k^{(r-1)}, \;F_n^{(0)} = F_n, \;F_0^{(r)} =0, \;F_1^{(r)} = 1,NEWLINE\]NEWLINE where \(n\) is a natural number and \(F_n\) is the \(n\)th Fibonacci number. Let NEWLINE\[NEWLINEA_{r,n} = \begin{pmatrix} F_{n}^{(r)} & F_{n+1}^{(r)} & \dots & F_{n+r+1}^{(r)} \\ F_{n+1}^{(r)} & F_{n+2}^{(r)} & \dots & F_{n+r+2}^{(r)} \\ \vdots & \vdots & \ddots & \vdots \\ F_{n+r+1}^{(r)} & F_{n+r+2}^{(r)} & \dots & F_{n+2r+2}^{(r)}\end{pmatrix}NEWLINE\]NEWLINE and NEWLINE\[NEWLINEQ_{r+2} = \begin{pmatrix} 0 & 1 & 0 & \dots & 0 & 0 \\ 0 & 0 & 1 & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & 1 & 0 \\ 0 & 0 & 0 & \dots & 0 & 1 \\ q_1 & q_2 & q_3 & \dots & q_{r+1} & q_{r+2} \end{pmatrix},NEWLINE\]NEWLINE where \(q_{r+2} = F_2^{(r)},\) \(q_{r+1} = F_3^{(r)} - F_2^{(r)}q_{r+2},\) \(q_{r} = F_4^{(r)} - F_3^{(r)}q_{r+2} - F_2^{(r)}q_{r+1},\) \(...,\) \(q_{1} = F_{r+3}^{(r)} - F_{r+2}^{(r)}q_{r+2} - ... - F_{2}^{(r)}q_{2}.\) It is proved (Theorem 2.2) that NEWLINE\[NEWLINEA_{r,n} = Q_{r+2}^nA_{r,0}NEWLINE\]NEWLINE and (Theorem 3.3) that for all natural numbers \(r\) and \(n\) NEWLINE\[NEWLINE\det(A_{r,n}) = (-1)^{n+\lfloor(r+3)/2 \rfloor}.NEWLINE\]