This licentiate 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.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:su-176124 |
| Date | January 2019 |
| Creators | NDIKUBWAYO, INNOCENT |
| Publisher | Stockholms universitet, Matematiska institutionen, Makerere University, Stockholm |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Licentiate thesis, comprehensive summary, info:eu-repo/semantics/masterThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
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