$q$-Binomial Identities Finder

TL;DR

This paper introduces a symbolic computation method that automatically transforms $q$-hypergeometric identities into $q$-binomial identities, re-establishes known formulas, discovers new identities, and provides proofs through an algorithmic approach.

Contribution

The paper presents a novel algorithmic method for transforming and discovering $q$-binomial identities from $q$-hypergeometric identities, including proof generation.

Findings

Re-derivation of known $q$-binomial identities like $q$-Saalsch"utz's formula

Discovery of numerous new $q$-binomial identities

Automated proof generation for the identities

Abstract

This paper presents a symbolic computation method for automatically transforming $q$-hypergeometric identities to $q$-binomial identities. Through this method, many previously proven $q$-binomial identities, including $q$-Saalsch\"utz's formula and $q$-Suranyi's formula, are re-fund, and numerous new ones are discovered. Moreover, the generation of the identities is accompanied by the corresponding proofs. During the transformation process, different ranges of variable values and various combinations of $q$-Pochhammer symbols yield different identities. The algorithm maps variable constraints to positive elements in an ordered vector space and employs a backtracking method to provide the feasible variable constraints and $q$-binomial coefficient combinations for each step.