[Paper Review] Searching for Strange Hypergeometric Identities By Sheer Brute Force

Moa Apagodu, Doron Zeilberger|ArXiv.org|Feb 26, 2008

Polynomial and algebraic computation1 references3 citations

TL;DR

This paper presents a brute-force computational approach to discover new hypergeometric identities by testing whether terminating hypergeometric series $ F(-an, bn+b_0, cn+c_0, x) $ yield hypergeometric sequences with rational ratio functions. The method identifies two new infinite families of closed-form evaluations, including a perturbed Kummer-type identity at $ x = -1 $, verified via Zeilberger’s algorithm and symbolic computation tools in the Maple package BruteTwoFone.

ABSTRACT

We describe a systematic search for all strange hypergeometric identities up to a certain complexity with sheer brute force that lead us to the discovery of two new infinite families of closed-form evaluations.

Motivation & Objective

• To systematically search for previously unknown hypergeometric identities with closed-form evaluations using computational brute force.
• To overcome limitations of prior methods that only detect identities yielding first-order recurrences via Zeilberger’s algorithm.
• To identify infinite families of exact evaluations for terminating hypergeometric series $ F(-an, bn+b_0, cn+c_0, x) $, particularly at special values of $ x $.
• To develop and deploy a computational framework using the Maple package BruteTwoFone to automate the discovery and verification of such identities.
• To distinguish novel identities from known ones (e.g., Chu-Vandermonde, Kummer) through systematic filtering and symbolic elimination.

Proposed method

• Fix parameters $ a, b, c, b_0, c_0 $ and search for values of $ x $ such that the ratio $ r_n = u_{n+1}/u_n $ is a rational function of degree $ \\-leq d $, where $ u_n = F(-an, bn+b_0, cn+c_0, x) $.
• Use numerical evaluation at $ n = 0, 1, ..., 2d+6 $ to generate a system of linear equations in the coefficients of the rational function $ r_n = P(n)/Q(n) $, with $ P, Q $ of degree $ d $.
• Apply linear algebra to test solvability of the system, and use the greatest common divisor of two resultants (polynomials in $ x $) to identify candidate $ x $-values that yield hypergeometric sequences.
• Leverage the Maple package BruteTwoFone, particularly procedures `NakhD` and `NakhDx`, to automate the search and detection of rational ratio functions.
• Verify conjectured identities rigorously using Zeilberger’s algorithm to prove that the resulting hypergeometric sequence satisfies a linear recurrence.
• Use transformation formulas and contiguous relations to identify equivalent identities and filter out known cases (e.g., Gauss, Kummer, Pfaff-Saalschütz).

Experimental results

Research questions

• RQ1 Can a brute-force computational search detect hypergeometric identities that evade traditional Zeilberger-based methods due to higher-order recurrences?
• RQ2 Are there new infinite families of closed-form evaluations for terminating hypergeometric series $ F(-an, bn+b_0, cn+c_0, x) $ at special values of $ x $?
• RQ3 Can the method systematically identify and distinguish novel identities from known ones such as Chu-Vandermonde or Kummer’s evaluations?
• RQ4 Is there a uniform parametric form for closed-form evaluations beyond the known families, particularly for $ x = -3 $ and other special points?
• RQ5 Can symbolic computation tools like the Maple package BruteTwoFone automate the discovery and verification of such identities at scale?

Key findings

• The authors discovered a new infinite family of closed-form evaluations at $ x = -1 $, given by Theorem 1, which generalizes Kummer’s identity and includes perturbations of the classical hypergeometric function.
• For $ a = 2 $, $ b $ arbitrary, $ c = -2n + 2r - b $, and $ x = -1 $, the identity $ F(-2n, b, -2n + 2r - b, -1) $ evaluates to a ratio of Pochhammer symbols times a polynomial in $ n $ of degree $ r-1 $, with explicit coefficients.
• The method successfully identified $ x = (1 /pm √{3}i)/2 $ as a candidate for closed-form evaluation when $ a=1 $, $ b=-3 $, $ c=-2 $, $ b_0=-1 $, $ c_0=0 $, matching known results from prior work.
• After filtering known identities, 19 novel, inequivalent hypergeometric identities remained, most of which were already documented in earlier work and are accessible via the package’s precomputed database.
• Conjecture 1 posits that $ F(-2n, -1/2 + i, -3n - 1/2 + j, -3) $ admits closed-form evaluation for all integers $ i, j $, though no uniform expression in $ i $ and $ j $ has been found.
• The computational pipeline, implemented in the Maple package BruteTwoFone, enables automated discovery and verification of such identities using rational function ratio testing and GCD-based filtering of candidate $ x $-values.