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Innocent Ndikubwayo: Topics in polynomial sequences defined by linear recurrences

Time: Tue 2019-12-17 10.00

Location: Kräftriket, house 5, room 14

Subject area: Complex Variables

Doctoral student: Innocent Ndikubwayo , Stockholms universitet

Opponent: Tamas Forgacs, California State University, Fresno

Supervisor: Boris Shapiro, Stockholms universitet

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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.