On Kadell's two Conjectures for the $q$-Dyson Product
Yue Zhou
TL;DR
This paper extends existing formulas of the $q$-Dyson product to prove Kadell's conjecture and its $q$-analog, clarifying the conjecture's validity and limitations.
Contribution
It introduces extended formulas for the $q$-Dyson product, proving Kadell's conjecture and establishing a $q$-analog, advancing understanding of these mathematical structures.
Findings
Kadell's conjecture for the Dyson product is proven.
The error in Kadell's $q$-analogous conjecture is identified.
A $q$-analog of Kadell's conjecture is established.
Abstract
By extending Lv-Xin-Zhou's first layer formulas of the -Dyson product, we prove Kadell's conjecture for the Dyson product and show the error of his -analogous conjecture. With the extended formulas we establish a -analog of Kadell's conjecture for the Dyson product.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Combinatorial Mathematics · Advanced Algebra and Geometry