(\(\pm 1\))-invariant sequences and truncated Fibonacci sequences
Let \(P\) and \(D\) denote the Pascal matrix \(\bigl[\binom{i}{j}\bigr]\), (\(i,j=0,1,2,\dots\)) and the diagonal matrix \(\text{diag}((-1)^0,(-1)^1,(-1)^2,\dots)\), respectively. An infinite-dimensional real vector \(\mathbf x\) is called a \(\lambda\)-invariant sequence if \(PD\mathbf x=\lambda\mathbf x\). It is known that \(1\) and \(-1\) are the only eigenvalues of \(PD\). Various kinds of \((\pm 1)\)-invariant sequences are introduced by \textit{Z. -H. Sun} [Fibonacci Q. 39, 324--333 (2001; Zbl 0987.05013)]. In this paper the authors find some characterization of \((\pm 1)\)-invariant sequences in connection with the matrices \(P\) and \(Q\). They also introduce truncated Fibonacci and Lucas sequences and proves that an infinite dimensional real vector \(\mathbf x\) is \((-1)\)-invariant (\(1\)-invariant respectively) if and only if \(\mathbf x\) is expressible as a linear combination of truncated Fibonacci (truncated Lucas respectively) sequences.
scientific article; zbMATH DE number 2137803