Advanced Computer Algebra for Determinants

TL;DR

This paper proves three longstanding conjectures on determinant evaluations related to plane partitions and rhombus tilings using advanced computer algebra techniques, extending Zeilberger's holonomic ansatz.

Contribution

It introduces enhanced computer algebra methods to prove complex determinant conjectures, including one posed by Andrews in 1980.

Findings

Proof of three determinant conjectures

Extension of Zeilberger's holonomic ansatz

A new challenge conjecture on Andrews's determinant

Abstract

We prove three conjectures concerning the evaluation of determinants, which are related to the counting of plane partitions and rhombus tilings. One of them was posed by George Andrews in 1980, the other two were by Guoce Xin and Christian Krattenthaler. Our proofs employ computer algebra methods, namely, the holonomic ansatz proposed by Doron Zeilberger and variations thereof. These variations make Zeilberger's original approach even more powerful and allow for addressing a wider variety of determinants. Finally, we present, as a challenge problem, a conjecture about a closed-form evaluation of Andrews's determinant.