Convergence radii for eigenvalues of tri-diagonal matrices (Q2655355)

The authors consider a family of infinite tri-diagonal matrices of the form \(L+zB\), where \(L\) is a diagonal and \(B\) is an off-diagonal matrix with specific entries. Under some conditions which include a parameter \(a\), \(0\leq a<2\), its spectrum is discrete and its eigenvalues are analytic functions. Let \(a_{k}(n)\) and \(R_{n}\) be the coefficients and the convergence radius, respectively, of the Taylor series of these functions. They give upper bounds for \(a_{k}(n)\) and, using them, they also give an upper bound for \(R_{n}\) if \(0\leq a<11/6\).