On Zeilberger's Constant Term for Andrews' TSSCPP Theorem

TL;DR

This paper provides a simplified proof for Zeilberger's constant term identities, reducing complex multivariable constants to single determinants, and extends these results to new classes of matrices and root systems.

Contribution

It introduces a reduction technique for Zeilberger's constant terms, converting them into single determinants, and applies this to solve a key problem and extend to root system BC.

Findings

Zeilberger's constant terms are expressed as single determinants.

A reduction from 2k to k variables in constant terms is achieved.

The approach extends to sum of minors of certain matrices and root systems.

Abstract

This paper studies Zeilberger's two prized constant term identities. For one of the identities, Zeilberger asked for a simple proof that may give rise to a simple proof of Andrews theorem for the number of totally symmetric self complementary plane partitions. We obtain an identity reducing a constant term in $2k$ variables to a constant term in $k$ variables. As applications, Zeilberger's constant terms are converted to single determinants. The result extends for two classes of matrices, the sum of all of whose full rank minors is converted to a single determinant. One of the prized constant term problems is solved, and we give a seemingly new approach to Macdonald's constant term for root system of type BC.