Computation with hyperexponential functions

Abstract

A multivariate hyperexponential function is a function whose"logarithmic derivatives" are rational. Examples ofhyperexponential functions include rational functions, exponentialfunctions, and hypergeometric terms. Hyperexponential functionsplay an important role in the handling of analytic andcombinatorial objects. We present a few algorithms applicable tothe manipulation of hyperexponential functions in an uniformway. Let F be a field of characteristic zero, onwhich derivation operatorsδ1,...,δℓand difference operators (automorphisms)σℓ+1,...,σm act. Let Ebe an F-algebra. Assume that theδi for 1≤ i ≤ ℓ andσj forℓ + 1 ≤ m can be extended toE as derivation and difference operators.Moreover, these operators commute with each other onE. A hyperexponential element ofE over F is defined to be anonzero element h ∈E such that δ1(h) =r1h,...,δℓ(h)=rℓh,σℓ+1(h)=rℓ+1h,...,σm(h)=rmh for some r1,...,rm ∈F. These rational functions are called(rational) certificates for h.