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Rational reductions for holonomic sequences

Rong-Hua Wang

Abstract

Given a holonomic sequence $F(n)$, we characterize rational functions $r(n)$ so that $r(n)F(n)$ can be summable. We provide upper and lower bounds on the degree of the numerator of $r(k)$ and show the denominator of $r(n)$ can be read from annihilators of $F(k)$. This illustration provides the so-called rational reductions which can be used to generate new multi-sum equalities and congruences from known ones.

Rational reductions for holonomic sequences

Abstract

Given a holonomic sequence , we characterize rational functions so that can be summable. We provide upper and lower bounds on the degree of the numerator of and show the denominator of can be read from annihilators of . This illustration provides the so-called rational reductions which can be used to generate new multi-sum equalities and congruences from known ones.
Paper Structure (4 sections, 10 theorems, 61 equations)

This paper contains 4 sections, 10 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Summablity of $p(n)F(n)$
  3. Summablity of $r(n)F(n)$
  4. Applications of rational reductions

Key Result

Theorem 2.1

Suppose $F(n)$ is a holonomic sequence and $L=\sum\limits_{i=0}^{J}a_i(n)\sigma^i\in {\mathrm{ann\space}} F(n)$. Then there exists a nonzero polynomial $p(n)$ such that $p(n)F(n)$ is summable and where $d=\deg L$ and $C_L$ is the continued zero index of $L$.

Theorems & Definitions (14)

  • Theorem 2.1
  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Example 2.7
  • Theorem 3.1
  • Lemma 3.2
  • Example 3.3
  • ...and 4 more