College of Arts and Sciences Third Order a - Hypergeometric Functions

To solve globally bounded order 3 linear differential equations with rational function coefficients, this thesis introduces a partial 3F2-solver (Section 3.2) and F1-solver (Chapter 4) where 3F2 is the hypergeometric function 3F2(a1, a2, a3; b1, b2 |x) and F1 is the Appell’s F1(a, b1, b2, c |x, y). To investigate the relations among order 3 multivariate hypergeometric functions, this thesis presents two multivariate tools: compute homomorphisms (Algorithm 5.3.10) of two D-modules where D is a multivariate differential ring, and compute projective homomorphisms (Algorithm 5.4.5) using the tensor product module and Algorithm 5.3.10. As an application, all irreducible order 2 subsystems from reducible order 3 systems turn out to come from Gauss hypergeometric function 2F1(a, b; c |x) (Chapter 6).