Title of Thesis:

Topics in polynomial sequences defined by linear recurrences

Author of Thesis:

Innocent Ndikubwayo

External reviewer:

Tamas Forgacs (Fresno State)

Supervisor:

Boris Shapiro

Abstract:

This licentiate thesis consists of two papers treating polynomial sequences defined by linear recurrences.

In Paper I, we establish necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence $\{P_i\}$ generated by a three-term recurrence relation $P_i(x)+ Q_1(x)P_{i-1}(x) +Q_2(x) P_{i-2}(x)=0$ with the standard initial conditions $P_{0}(x)=1, P_{-1}(x)=0$, where $Q_1(x)$ and $Q_2(x)$ are arbitrary real polynomials.

In Paper II, we study the root distribution of a sequence of polynomials $\{P_n(z)\}$ with the rational generating function $$\sum_{n=0}^{\infty} P_n(z)t^n= \frac{1}{1+ B(z)t^\ell +A(z)t^k}$$ for $(k,\ell)=(3,2)$ and $(4,3)$, where $A(z)$ and $B(z)$ are arbitrary polynomials in $z$ with complex coefficients. We show that the roots of $P_n(z)$ which satisfy $A(z)B(z)\neq 0$ lie on a real algebraic curve which we describe explicitly.