EIGENPROBLEMS FOR TRIDIAGONAL 2-TOEPLITZ MATRICES AND QUADRATIC POLYNOMIAL-MAPPINGS

Given a system of monic orthogonal polynomials (MOPS) {P-n(x)}(n great er than or equal to 0), we characterize all the sequences of monic ort hogonal polynomials {Q(n)(x)}(n greater than or equal to 0) such that Q(1)(x) = x - b, Q(2n)(x) = P-n[pi(2)(X)], n = 0, 1, 2,..., where pi(2 ) is a fixed polynomial of degree exactly 2 and b is a fixed complex n umber. With an appropriate choice of the MOPS {P-n(x)}(n greater than or equal to 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. Cover. We also find the relations between the Jaco bi matrices corresponding to the MOPS {P-n(x)}(n greater than or equal to 0) and {Q(n)(x)}(n greater than or equal to 0). Finally, we show t hat if {P-n(x)}(n greater than or equal to 0) is a semiclassical ortho gonal polynomial sequence, then so is {Q(n)(x)}(n greater than or equa l to 0) and, in particular, we analyze the classical case in detail. ( C) Elsevier Science Inc., 1997.