On the structure of compatible rational functions.

A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the structure of compatible rational functions. The theorem enables us to decompose a solution of such a system as a product of a rational function, several symbolic powers, a hyperexponential function, a hypergeometric term, and a q-hypergeometric term. We outline an algorithm for computing this product, and discuss how to determine the algebraic dependence of hyperexponential-hypergeometric elements. This is joint work with Shaoshi Chen, Royong Feng, and Guofeng Fu.