Stavros Garoufalidis
Abstract: A holonomic (i.e., $D$-finite) 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 Zeilberger states that every multisum sequence is holonomic. We show that 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 side bonus, we define a class of holonomic $G$-functions that come from enumerative combinatorics that complement the holonomic $G$-functions that appear from geometry and from arithmetic. In a separate paper we discuss an efficient ansatz for computing the singularities of the holonomic $G$-functions that come from enumerative combinatorics.
Key words: $G$-functions, holonomic sequences, $D$-finite sequences, Zeilberger, hypergeometric terms, quasi-unipotent monodromy, asymptotic expansions.
Notes: 7 pages, no figures.