How to Extend Karolyi and Nagy's BRILLIANT Proof of the Zeilberger-Bressoud q-Dyson Theorem in order to Evaluate ANY Coefficient of the q-Dyson Product

TL;DR

This paper extends Karolyi and Nagy's proof of the Zeilberger-Bressoud q-Dyson theorem to evaluate any coefficient of the q-Dyson product, providing a method to compute these coefficients as rational functions times q-multinomial coefficients.

Contribution

It introduces an extension of the proof technique to evaluate arbitrary coefficients of the q-Dyson product, not just the constant term.

Findings

Any coefficient can be expressed as a rational function times a q-multinomial coefficient.

The algorithm provides a systematic way to evaluate these coefficients.

The method generalizes the original proof to a broader class of coefficients.

Abstract

We show how to extend the Karolyi-Nagy beautiful proof of the Zeilberger-Bressoud q-Dyson theorem, (first proved by Zeilberger and Bressoud in 1985, and originally conjectured by George Andrews in 1975), that states that the constant term of a certain Laurent polynomial equals the q-multinomial coefficient, how to evaluate any other specific coefficient. The algorithm implies that any such coefficient is always a certain rational function (that the algorithm finds) times the q-multinomial coefficient.