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Innocent Ndikubwayo: Topics in polynomial sequences defined by linear recurrences

Tid: Ti 2019-12-17 kl 10.00

Ämnesområde: Complex Variables

Respondent: Innocent Ndikubwayo , Stockholms universitet

Opponent: Tamas Forgacs, California State University, Fresno

Handledare: Boris Shapiro, Stockholms universitet

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.

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