Eigenproblems for tridiagonal 2-Toeplitz matrices and quadratic polynomial mappings Original Research Article

Given a system of monic orthogonal polynomials (MOPS) {Pn(x)}n greater-or-equal, slanted 0, we characterize all the sequences of monic orthogonal polynomials {Qn(x)}n greater-or-equal, slanted 0 such that Q1(x) = x − b, Q2n(x) = Pn[π2(x)], n = 0, 1, 2, …, where π2 is a fixed polynomial of degree exactly 2 and b is a fixed complex number. With an appropriate choice of the MOPS {Pn(x)}n greater-or-equal, slanted 0, our results enables us to solve the eigenproblem of a tridiagonal 2-Toeplitz matrix, giving an alternative proof to a recent result by M. J. C. Gover. We also find the relations between the Jacobi matrices corresponding to the MOPS {Pn(x)}n greater-or-equal, slanted 0 and {Qn(x)}n greater-or-equal, slanted 0. Finally, we show that if {Pn(x)}n greater-or-equal, slanted 0 is a semiclassical orthogonal polynomial sequence, then so is {Qn(x)}n greater-or-equal, slanted 0, and, in particular, we analyze the classical case in detail.