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### Algorithms for Solving Linear Differential Equations with Rational Function Coefficients

 Title: Name(s): Algorithms for Solving Linear Differential Equations with Rational Function Coefficients. 47 views 10 downloads Imamoglu, Erdal, authorvan Hoeij, Mark, professor directing dissertationvan Engelen, Robert, university representativeAgashe, Amod S. (Amod Sadanand), committee memberAldrovandi, Ettore, committee memberAluffi, Paolo, 1960-, committee member Florida State University, degree granting institution College of Arts and Sciences, degree granting college Department of Mathematics, degree granting department text TextDoctoral Thesis monographic 2017 Florida State University Tallahassee, Florida computeronline resource 1 online resource (70 pages) English 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. FSU_SUMMER2017_Imamoglu_fsu_0071E_13942 (IID) 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. Mathematics http://purl.flvc.org/fsu/fd/FSU_SUMMER2017_Imamoglu_fsu_0071E_13942 FSU

Imamoglu, E. (2017). Algorithms for Solving Linear Differential Equations with Rational Function Coefficients. Retrieved from http://purl.flvc.org/fsu/fd/FSU_SUMMER2017_Imamoglu_fsu_0071E_13942