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On the Summability Problem of Multivariate Rational Functions in the Mixed Case

Shaoshi Chen, Lixin Du, Hanqian Fang, Yisen Wang

TL;DR

The paper addresses the summability problem for multivariate rational functions under mixed shift and $q$-shift operators. It develops a framework based on orbital decompositions, isotropy groups, and difference transformations to reduce general problems to simple fractions and to lower-dimensional subproblems. A complete solution for the rational case is achieved, including a rank-based criterion involving $G_d/H_d$ and a rank-1 theorem that decomposes the numerator into sums of discrete differences, illustrated by explicit $q$-summability examples and telescoping representations. The approach enables algorithmic symbolic summation for multivariate functions and lays groundwork for extending Gosper-style methods and creative telescoping to multivariate hypergeometric terms and multi-sum identities.

Abstract

Continuing previous work, this paper focuses on the summability problem of multivariate rational functions in the mixed case in which both shift and $q$-shift operators can appear. Our summability criteria rely on three ingredients including orbital decompositions, Sato's isotropy groups, and difference transformations. This work settles the rational case of the long-term project aimed at developing algorithms for symbolic summation of multivariate functions.

On the Summability Problem of Multivariate Rational Functions in the Mixed Case

TL;DR

The paper addresses the summability problem for multivariate rational functions under mixed shift and -shift operators. It develops a framework based on orbital decompositions, isotropy groups, and difference transformations to reduce general problems to simple fractions and to lower-dimensional subproblems. A complete solution for the rational case is achieved, including a rank-based criterion involving and a rank-1 theorem that decomposes the numerator into sums of discrete differences, illustrated by explicit -summability examples and telescoping representations. The approach enables algorithmic symbolic summation for multivariate functions and lays groundwork for extending Gosper-style methods and creative telescoping to multivariate hypergeometric terms and multi-sum identities.

Abstract

Continuing previous work, this paper focuses on the summability problem of multivariate rational functions in the mixed case in which both shift and -shift operators can appear. Our summability criteria rely on three ingredients including orbital decompositions, Sato's isotropy groups, and difference transformations. This work settles the rational case of the long-term project aimed at developing algorithms for symbolic summation of multivariate functions.
Paper Structure (10 sections, 15 theorems, 67 equations)

This paper contains 10 sections, 15 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Additive decompositions
  4. Orbital decompositions
  5. Reducing to simple fractions
  6. Isotropy groups
  7. Summability criteria
  8. Difference transformations
  9. Examples for applications
  10. Conclusion and future work

Key Result

Lemma 3.1

If $f\in V_{[d]_G,j}$ and $P\in {\mathbb{F}}( {\hat{\bf x}}_1)[G]$, then $P(f)\in V_{[d]_G,j}$.

Theorems & Definitions (33)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Proposition 3.4
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • ...and 23 more