AF: Small: A-Hypergeometric Solutions of Linear Differential Equations

Florida State University, Tallahassee FL

Investigators

Abstract

Many scientific laws are captured as differential equations -- formulas that relates a quantity (such as position) to its derivatives (velocity and acceleration). Differential equations are common in science, mathematics and engineering. They can be solved numerically, but may also have a closed-form solution -- an exact solution written in terms of familiar functions. Except for small equations, closed-form solutions were thought to be rare. Algorithms for second-order equations, which were developed by the PI and his students, have demonstrated that closed form solutions of linear differential equations with polynomial coefficients are actually common; solutions can often be written in terms of Gauss's hypergeometric function, a familiar function in differential equations. This project aims to find out if closed form solutions are also common for higher order equations. The PI will work with two graduate students to develop algorithms to search for solutions in terms of A-hypergeometric functions, which were introduced by Gel'fand, Kapranov and Zelevinski. There is a wide variety of A-hypergeometric functions, which correspond to polytopes. A-hypergeometric functions can be multivariate, so several tools that currently only exist for univariate equations need to be generalized. A key motivating question is if convergent integer-series solutions can always be written in terms of A-hypergeometric functions. If true, then A-hypergeometric solutions would be common in areas of mathematics and science that involve combinatorial structures or integrals, such as the Ising model or Feynman diagrams in physics. This question leads to several others on A-hypergeometric functions, some of which can be settled by the algorithms to be developed in this project.