[Paper Review] A conjecture in the problem of rational definite summation

Mark van Hoeij|ArXiv.org|Oct 10, 2002

semigroups and automata theory1 references3 citations

TL;DR

This paper proposes a conjecture that reduces the problem of determining whether the definite sum of a rational function over k from 0 to n is rational to checking the same property for individual equivalence classes of partial fraction terms. If true, the conjecture enables a systematic algorithmic approach to rational definite summation by exploiting structural symmetries in the denominator factors.

ABSTRACT

A conjecture is given that, if true, could lead to an algorithm for computing definite sums of rational functions.

Motivation & Objective

• To address the fundamental problem of deciding whether the definite sum ∑ₖ₌₀ⁿ F(n,k) of a rational function F(n,k) is itself a rational function of n.
• To reduce the general summation problem to simpler subproblems by grouping partial fraction terms into equivalence classes based on algebraic symmetry.
• To provide a theoretical foundation for an algorithmic approach to rational definite summation by identifying structural conditions under which sums remain rational.
• To explore the role of automorphisms and symmetry types (generic, symmetric, rational) in enabling term cancellations that preserve rationality of the sum.
• To investigate whether rationality of the full sum implies rationality of sums over each equivalence class, thereby enabling modular algorithm design.

Proposed method

• Define an equivalence relation ∼ₚ on irreducible polynomials in C(n)[k] via affine transformations ψₐ,ᵦ,𝒸(k) = ak + bn + c, identifying polynomials that are related by such maps up to units.
• Decompose a rational function F(n,k) into a polynomial part Q and partial fraction terms Fᵢ = Aᵢ/Bᵢ^{dᵢ}, with deg_k(Aᵢ) < deg_k(Bᵢ) and Bᵢ irreducible.
• Group partial fraction terms into equivalence classes Gᵢ based on equal dᵢ and Bᵢ ∼ₚ Bⱼ, forming sums over symmetric families of terms.
• Propose that if ∑ₖ₌₀ⁿ F(n,k) is rational, then each ∑ₖ₌₀ⁿ Gᵢ(n,k) must also be rational—this is the central conjecture.
• Classify irreducible polynomials B ∈ C(n)[k] by the size of their automorphism group Aut(B), distinguishing generic (1 element), symmetric (2 elements), and rational types (∞ elements).
• Use these symmetry types to guide the design of cancellation detection procedures, especially for rational-type terms where more complex cancellations occur.

Experimental results

Research questions

• RQ1 Under what conditions is the definite sum ∑ₖ₌₀ⁿ F(n,k) of a rational function F(n,k) a rational function of n?
• RQ2 Can the rationality of the full sum be determined by analyzing the sums over equivalence classes of partial fraction terms?
• RQ3 Do specific symmetry types (generic, symmetric, rational) in the denominator polynomials enable predictable cancellations that preserve rationality?
• RQ4 Is there a complete algorithmic procedure for rational definite summation that systematically detects all possible cancellations among equivalent partial fraction terms?
• RQ5 Does the existence of a linear recurrence with polynomial coefficients for R(n) = ∑ₖ₌₀ⁿ F(n,k) imply that R(n) is rational, under certain denominator constraints?

Key findings

• The conjecture posits that if ∑ₖ₌₀ⁿ F(n,k) is rational, then each sum ∑ₖ₌₀ⁿ Gᵢ(n,k) over an equivalence class of partial fraction terms must also be rational, reducing the global problem to smaller, structured subproblems.
• The paper identifies three symmetry types—generic, symmetric, and rational—based on the automorphism group of irreducible denominator factors, which determine the complexity of possible cancellations.
• Examples show that nontrivial cancellations can occur even in the generic type (e.g., F = 1/B(n,k) − 1/B(n,n−k)), where the sum is zero despite individual terms being nonzero.
• For rational-type terms, more complex cancellations are possible, such as in 1/(k+1) + 2/(k+2+n) − 2/(2k+1), which sums to zero after rewriting with symmetric terms.
• The sum R(n) = ∑ₖ₌₀ⁿ 1/(k + n(n+1)/2) does not satisfy any linear recurrence with polynomial coefficients over C[n], implying it is not a rational function.
• The failure of such a recurrence is proven via contradiction using prime density arguments and properties of disjoint sets Sₙ, showing that the denominator growth rate contradicts the prime number theorem if a recurrence existed.