Computation with hyperexponential functions

Abstract

A multivariate hyperexponential function is a function whose "logarithmic derivatives" are rational. Examples of hyperexponential functions include rational functions, exponential functions, and hypergeometric terms. Hyperexponential functions play an important role in the handling of analytic and combinatorial objects. We present a few algorithms applicable to the manipulation of hyperexponential functions in an uniform way. Let <i>F</i> be a field of characteristic zero, on which derivation operators δ<inf>1</inf>,...,δ<inf>ℓ</inf> and difference operators (automorphisms) σ<inf>ℓ+1</inf>,..., σ<inf>m</inf> act. Let <i>E</i> be an <i>F</i>-algebra. Assume that the δ<inf><i>i</i></inf> for 1 ≤ <i>i</i> ≤ ℓ and σ<inf><i>j</i></inf> for ℓ + 1 ≤ <i>m</i> can be extended to <i>E</i> as derivation and difference operators. Moreover, these operators commute with each other on <i>E.</i> A hyperexponential element of <i>E</i> over <i>F</i> is defined to be a nonzero element <i>h</i> ∈ <i>E</i> such that δ<inf>1</inf>(<i>h</i>) = <i>r</i><inf>1</inf><i>h</i>, ...,δ<inf>ℓ</inf>(<i>h</i>) = <i>r</i><inf>ℓ</inf><i>h</i>, σ<inf>ℓ+1</inf>(<i>h</i>) = <i>r</i><inf>ℓ+1</inf><i>h</i>,...,σ<inf><i>m</i></inf>(<i>h</i>) = <i>r</i><inf><i>m</i></inf><i>h</i> for some <i>r</i><inf>1</inf>,..., <i>r<inf>m</inf></i> ∈ <i>F</i>. These rational functions are called (rational) certificates for <i>h.</i>