Polynomial Sequences Generated by Linear Recurrences : Location and Reality of Zeros

In this thesis, we study the problem of location of the zeros of individual polynomials in sequences of polynomials generated by linear recurrence relations. In paper I, we establish the necessary and sufficient conditions that guarantee hyperbolicity of all the polynomials generated by a three-term recurrence of length 2, whose coefficients are arbitrary real polynomials. These zeros are dense on the real intervals of an explicitly defined real semialgebraic curve. Paper II extends Paper I to three-term recurrences of length greater than 2. We prove that there always exist non-hyperbolic polynomial(s) in the generated sequence. We further show that with at most finitely many known exceptions, all the zeros of all the polynomials generated by the recurrence lie and are dense on an explicitly defined real semialgebraic curve which consists of real intervals and non-real segments. The boundary points of this curve form a subset of zero locus of the discriminant of the characteristic polynomial of the recurrence. Paper III discusses the zero set for polynomials generated by three-term recurrences of lengths 3 and 4 with arbitrary polynomial coefficients. We prove that except the zeros of the polynomial coefficients, all the zeros of every generated polynomial lie on an explicitly defined real semialgebraic curve. Paper IV extends the results in paper III and generalizes a conjecture by K. Tran [2]. We consider a three-term recurrence relation of any length whose coefficients are arbitrary complex polynomials and prove that with the exception of the zeros of the polynomial coefficients, all the zeros of every generated polynomial lie on a real algebraic curve. We derive the equation of this curve. Paper V establishes the necessary and sufficient conditions guaranteeing the reality of all the zeros of every polynomial generated by a special five-term recurrence with real coefficients. We put the problem in the context of banded Toeplitz matrices whose associated Laurent polynomial is holomorphic in the punctured plane. We interpret the conditions in terms of the positivity/negativity of the discriminant of a certain polynomial whose coefficients are explicit functions of the parameters in the recurrence