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Creative Telescoping for Hypergeometric Double Sums

Peter Paule, Carsten Schneider

TL;DR

The paper addresses computing $P$-finite recurrences for hypergeometric multi-sums $S(n)=\sum F(n,\dots)$ by integrating creative telescoping with contiguous-relations theory to obtain hook-type recurrences and a scalar rational solution of a parameterized recurrence.A practical double-sum method is developed: compute recurrences for the inner sum, craft a certificate $g(n,r)$, and telescope over the outer index to derive a recurrence for $S(n)$; the method is extended to accommodate multiple sums and refined for speed.The authors provide concrete instances (e.g., Blodgett–Andrews–Paule; Ahlgren–Rivoal–Krattenthaler), show how recurrences certify identities, and present substantial speedups via preprocessing, heuristic checks, and Gosper-Petkovšek-based numerator predictions.They also demonstrate that the framework generalizes to the $q$-hypergeometric setting and is implemented in Sigma, offering a scalable, automatable approach for a broad class of combinatorial and number-theoretic identities.

Abstract

We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which guarantees the applicability of our method for many input sums. In addition, we elaborate new techniques to optimize the underlying key task of our method to compute rational solutions of parameterized linear recurrences.

Creative Telescoping for Hypergeometric Double Sums

TL;DR

The paper addresses computing $P$-finite recurrences for hypergeometric multi-sums $S(n)=\sum F(n,\dots)$ by integrating creative telescoping with contiguous-relations theory to obtain hook-type recurrences and a scalar rational solution of a parameterized recurrence.A practical double-sum method is developed: compute recurrences for the inner sum, craft a certificate $g(n,r)$, and telescope over the outer index to derive a recurrence for $S(n)$; the method is extended to accommodate multiple sums and refined for speed.The authors provide concrete instances (e.g., Blodgett–Andrews–Paule; Ahlgren–Rivoal–Krattenthaler), show how recurrences certify identities, and present substantial speedups via preprocessing, heuristic checks, and Gosper-Petkovšek-based numerator predictions.They also demonstrate that the framework generalizes to the $q$-hypergeometric setting and is implemented in Sigma, offering a scalable, automatable approach for a broad class of combinatorial and number-theoretic identities.

Abstract

We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which guarantees the applicability of our method for many input sums. In addition, we elaborate new techniques to optimize the underlying key task of our method to compute rational solutions of parameterized linear recurrences.
Paper Structure (17 sections, 1 theorem, 121 equations, 4 tables)

This paper contains 17 sections, 1 theorem, 121 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Summation Methods for Single Sums
  3. A slight but important variation
  4. Existence of Recurrences
  5. The Double Sum Method
  6. More Flexibility in Specifying Hypergeometric Double Sums
  7. Further Examples
  8. Blodgett--Andrews--Paule Sum
  9. Ahlgren--Rivoal--Krattenthaler--Sum
  10. The Method Extended to Multiple Sums
  11. Speeding Up our Multi-Sum Method
  12. Preprocessing of the input sums
  13. Heuristic Check for the number of solutions
  14. Improvement I: Produce an optimal denominator and degree bound
  15. Improvement II: producing an optimal system
  16. ...and 2 more sections

Key Result

Theorem 1

Let $a_1,\dots,a_{q+1}$ and $b_1,\dots,b_q$ be complex parameters. For $0 \leq l \leq q$ let be pairwise different tuples with non-negative integer entries. Then there exist $c_0, \dots, c_q$ in $\mathbb{K}$, not all $0$, and a polynomial $C(x) \in \mathbb{K}[x]$ such that for all $k \geq 0$, Moreover, $C(0)=0$, and if $C(x) \neq 0$, for the polynomial degree of $C(x)$ one has

Theorems & Definitions (27)

  • Example 1
  • Example 2
  • Remark 1
  • Theorem 1: Theorem 1A in Paule:21
  • Example 3
  • Example 4: Cont.
  • Example 5: Cont.
  • Example 6: Cont.
  • Example 7: Cont.
  • Example 8
  • ...and 17 more