A note on the location of zeros of polynomials defined by linear recursions (Q2714424)

Let \(G_n(x)=P(x)G_{n-1}(x)+Q(x)G_{n-2}(x)\), shortly \(G_n(P(x),Q(x),G_0(x),G_1(x))\), be a second order polynomial sequence over \(\mathbb C\). A representation of \(G_n\) in the form of the determinant of a tridiagonal matrix \(A_n\) of order \(n\) enables the reduction of the problem of locating the zeros of the polynomials \(G_n\) to the problem of locating the eigenvalues of \(A_n\). Using Brauer's and Gershgorin's theorems for locating eigenvalues, two general results for locating the zeros of the polynomials \(G_n\) are proved. Using Gershgorin's theorems, the author proved recently the estimate \(|z|\leq\max\{|e|+|c|,2\}\) for polynomials \(G_n(x,1,c,1+e)\) with \(c,e\neq 0\in{\mathbb C}\) [Acta Acad. Paed. Agriensis, Sect. Math. 25, 15-20 (1998; Zbl 0923.11034) and Publ. Math. 55, 453-464 (1999; Zbl 0960.11007)]. In the paper it is also shown that the estimate based on Brauer's theorem generally leads to a better result for the zeros \(z\) of these polynomials \(\{z\in{\mathbb C}:|z+e|\cdot|z|\leq 2|z|\}\cup\{z\in{\mathbb C}:|z|\leq 2\}\) which can be further simplified when relations between \(c\) and \(e\) are known.