Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parallel Summation in P-Recursive Extensions

Shaoshi Chen, Ruyong Feng, Manuel Kauers, Xiuyun Li

TL;DR

Addresses the problem of extending parallel summation to indefinite sums whose summands are rational in the index and a $P$-recursive sequence within a difference-field framework. The approach models the summand in a multivariate rational function field $F=K(t_0,\dots,t_{n-1})$ with automorphism $\sigma$, and employs a two-step, Abramov-style strategy to separate the normal and special parts of the denominator using dispersions and $\sigma$-equivalence to constrain possibilities. Key contributions include (i) an algebraic framework for denominators via local/global dispersions, (ii) structural results and bounds on irreducible special polynomials in the $P$-recursive setting, and (iii) a practical method to decide $\sigma$-equivalence of irreducible polynomials, plus a connection to the C-finite case via diagonalizability when no new constants are present. In the C-finite regime, the work aligns with and reduces to known rational-solution algorithms, while highlighting open challenges in bounding multiplicities of special factors. Overall, the paper extends parallel summation to a broader class of sums with parameters, providing theoretical bounds and algorithmic steps toward partial decision procedures.

Abstract

We propose investigating a summation analog of the paradigm for parallel integration. We make some first steps towards an indefinite summation method applicable to summands that rationally depend on the summation index and a P-recursive sequence and its shifts. There is a distinction between so-called normal and so-called special polynomials. Under the assumption that the corresponding difference field has no unnatural constants, we are able to predict the normal polynomials appearing in the denominator of a potential closed form. We can also handle the numerator. Our method is incomplete so far as we cannot predict the special polynomials appearing in the denominator. However, we do have some structural results about special polynomials for the setting under consideration.

Parallel Summation in P-Recursive Extensions

TL;DR

Addresses the problem of extending parallel summation to indefinite sums whose summands are rational in the index and a -recursive sequence within a difference-field framework. The approach models the summand in a multivariate rational function field with automorphism , and employs a two-step, Abramov-style strategy to separate the normal and special parts of the denominator using dispersions and -equivalence to constrain possibilities. Key contributions include (i) an algebraic framework for denominators via local/global dispersions, (ii) structural results and bounds on irreducible special polynomials in the -recursive setting, and (iii) a practical method to decide -equivalence of irreducible polynomials, plus a connection to the C-finite case via diagonalizability when no new constants are present. In the C-finite regime, the work aligns with and reduces to known rational-solution algorithms, while highlighting open challenges in bounding multiplicities of special factors. Overall, the paper extends parallel summation to a broader class of sums with parameters, providing theoretical bounds and algorithmic steps toward partial decision procedures.

Abstract

We propose investigating a summation analog of the paradigm for parallel integration. We make some first steps towards an indefinite summation method applicable to summands that rationally depend on the summation index and a P-recursive sequence and its shifts. There is a distinction between so-called normal and so-called special polynomials. Under the assumption that the corresponding difference field has no unnatural constants, we are able to predict the normal polynomials appearing in the denominator of a potential closed form. We can also handle the numerator. Our method is incomplete so far as we cannot predict the special polynomials appearing in the denominator. However, we do have some structural results about special polynomials for the setting under consideration.
Paper Structure (8 sections, 13 theorems, 52 equations, 1 algorithm)

This paper contains 8 sections, 13 theorems, 52 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Parallel summation
  3. Local dispersions and denominator bounds
  4. Number of irreducible special polynomials
  5. The P-recursive case
  6. Degrees of irreducible special polynomials
  7. The $\sigma$-Equivalence Problem
  8. The C-finite case

Key Result

Lemma 7

Let $f=a/b \in F$ with $a, b\in R$ and $\gcd(a, b)=1$ and let $p \in R$ be an irreducible normal factor of $b$. Then $\operatorname{disp}_p(\Delta(f)) = \operatorname{disp}_p(f) + 1$ and $\operatorname{disp}(\Delta(f)) = \operatorname{disp}(f) + 1$.

Theorems & Definitions (29)

  • Example 3
  • Example 4
  • Definition 6
  • Lemma 7
  • proof
  • Theorem 8
  • proof
  • Example 9
  • Lemma 10
  • Corollary 11
  • ...and 19 more