G-functions and Multisum versus Holonomic Sequences

A holonomic (i.e., D-finite, or P -recursive) sequence is one that satisfies a linear recursion relation with polynomial coefficients. A multisum sequence is one that is given by a multisum of a proper hypergeometric term. A fundamental theorem of Wilf-Zeilberger states that every multisum sequence is holonomic. For over 15 years, it was accepted as a reasonable conjecture that the converse holds. Our main result is to prove and to explain why the converse does not hold, i.e., that there exist plenty holonomic sequences that are not balanced multisums. Our proof uses G-function theory and the quasi-unipotence of the local monodromy around the singularities. As a companion of our proof, we construct a class of G-functions that come from enumerative combinatorics that complement the G-functions that appear in geometry and arithmetic. In a separate paper we discuss an efficient ansatz for computing the singularities of the G-functions that come from enumerative combinatorics.