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Topics in polynomial sequences defined by linear recurrences
Stockholm University, Faculty of Science, Department of Mathematics. Makerere University. (Analysis)
2019 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Stockholm: Stockholm University, 2019.
Keywords [en]
recurrence relation, q-discriminant, generating function, polynomial sequence, support, real zeros
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-176124OAI: oai:DiVA.org:su-176124DiVA, id: diva2:1372350
Presentation
2019-12-17, 14, Kräftriket, house 5, Stockholm, 10:00 (English)
Opponent
Supervisors
Funder
Sida - Swedish International Development Cooperation Agency, 316Available from: 2019-11-22 Created: 2019-11-22 Last updated: 2019-11-22Bibliographically approved
List of papers
1. Around a Conjecture of K. Tran
Open this publication in new window or tab >>Around a Conjecture of K. Tran
(English)Manuscript (preprint) (Other academic)
Keywords
recurrence relation, q-discriminant, generating function
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-176123 (URN)
Funder
Sida - Swedish International Development Cooperation Agency, 316
Available from: 2019-11-22 Created: 2019-11-22 Last updated: 2019-11-22Bibliographically approved
2. CRITERION OF REALITY OF ZEROS IN A POLYNOMIAL SEQUENCE SATISFYING A THREE-TERM RECURRENCE RELATION
Open this publication in new window or tab >>CRITERION OF REALITY OF ZEROS IN A POLYNOMIAL SEQUENCE SATISFYING A THREE-TERM RECURRENCE RELATION
(English)Manuscript (preprint) (Other academic)
Abstract [en]

This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence {Pi } 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.

National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-162702 (URN)
Available from: 2018-12-07 Created: 2018-12-07 Last updated: 2019-11-22Bibliographically approved

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