The Structure of Multivariate Hypergeometric Terms

The structure of multivariate hypergeometric terms is studied. This leads to a proof of (a special case of) a conjecture formulated by Wilf and Zeilberger in 1992. A function u(n1, . . . , nd) with values in a field K is called a hypergeometric term if there exist rational functions Ri ∈ K(n1, . . . , nd), i = 1, . . . , d, such that u is solution of a system of d firstorder recurrences Si · u = Ri(n1, . . . , nd)u, i = 1, . . . , d, where Si denotes the shift operator with respect to ni (e.g., S1 · u(n1, . . . , nd) = u(n1 + 1, n2, . . . , nd)). In the univariate case, the numerator and denominator of R1 factor into linear factors over the algebraic closure K of K. This factorization induces an explicit form for univariate hypergeometric terms as Cρn ∏I i=1 (n+ αi) ki , where C is a constant, ρ ∈ K, αi ∈ K, ki ∈ {1,−1}, and I is the sum of the degrees of the numerator and denominator of R1. These terms thus express the Taylor coefficients of generalized hypergeometric series, whence their name. In the multivariate case, no such simple factorization exists, but the rational functions are related through the identities Rj(SjRi) = Ri(SiRj), 1 ≤ i, j ≤ d. A non-obvious consequence of these relations is the following theorem from an entirely elementary Appendix of [5] (see also [2]). The bivariate case was proved by Ore in [4]. Theorem 1 (Ore–Sato). Hypergeometric terms can be written (1) R(n1, . . . , nd) d ∏