This thesis introduces two new algorithms to find hypergeometric solutions of second order regular singular differential operators with rational function or polynomial coefficients. Algorithm 3.2.1 searches for solutions of type: exp(∫ r dx) ⋅ ₂F₁ (a₁,a₂;b₁;f) and Algorithm 5.2.1 searches for solutions of type exp(∫ r dx) (r₀ ⋅ ₂F₁(a₁,a₂;b₁;f) + r₁ ⋅ ₂F´₁ (a₁,a₂;b₁;f)) where f, r, r₀, r₁ ∈ ℚ̅(̅x̅)̅ and a₁,a₂,b₁ ∈ ℚ and denotes the Gauss hypergeometric function. The algorithms use modular reduction, Hensel lifting, rational function reconstruction, and rational number reconstruction to do so. Numerous examples from different branches of science (mostly from combinatorics and physics) showed that the algorithms presented in this thesis are very effective. Presently, Algorithm 5.2.1 is the most general algorithm in the literature to find hypergeometric solutions of such operators. This thesis also introduces a fast algorithm (Algorithm 4.2.3) to find integral bases for arbitrary order regular singular differential operators with rational function or polynomial coefficients. A normalized (Algorithm 4.3.1) integral basis for a differential operator provides us transformations that convert the differential operator to its standard forms (Algorithm 5.1.1) which are easier to solve. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2017. / May 25, 2017. / Differential Operators, Hypergeometric Solutions, Integral Bases / Includes bibliographical references. / Mark van Hoeij, Professor Directing Dissertation; Robert A. van Engelen, University Representative; Amod S. Agashe, Committee Member; Ettore Aldrovandi, Committee Member; Paolo B. Aluffi, Committee Member.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_552080 |
| Contributors | Imamoglu, Erdal (authoraut), van Hoeij, Mark (professor directing dissertation), van Engelen, Robert (university representative), Agashe, Amod S. (Amod Sadanand) (committee member), Aldrovandi, Ettore (committee member), Aluffi, Paolo, 1960- (committee member), Florida State University (degree granting institution), College of Arts and Sciences (degree granting college), Department of Mathematics (degree granting departmentdgg) |
| Publisher | Florida State University |
| Source Sets | Florida State University |
| Language | English, English |
| Detected Language | English |
| Type | Text, text, doctoral thesis |
| Format | 1 online resource (70 pages), computer, application/pdf |
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