Fibonacci polynomials and Sylvester determinant of tridiagonal matrix (Q969174)

Consider the \((n+1)\times(n+1)\) tridiagonal matrix \(M_{n}(x,y)\) with diagonal \(x,x+y,\dots,x+ny\), subdiagonal \(n,n-1,\dots,1\) and superdiagonal \(1,2,\dots,n\). The author proves that the eigenvalues of \(M_{n}(0,y)\) are \(\frac{1}{2}ny+\frac {1}{2}(n-2k)\sqrt{4+y^{2}}\) for \(k=0,1,\dots,n\) and hence that \(\det M_{n}(x,y)\) equals \( {\prod_{k=0}^{n}} \left( x+\frac{1}{2}ny+\frac{1}{2}(n-2k)\sqrt{4+y^{2}}\right) \). This is a generalization of an identity due to Sylvester (the case \(y=0\)). Generalized Fibonacci sequences with the recurrence relation \(f_{k+1}=yf_{k}+f_{k-1}\) arise in the reduction step from \(M_{n}(0,y)\) to \(M_{n-2}(0,y)\).