Proof of the Kresch-Tamvakis Conjecture

TL;DR

This paper proves a conjecture by Kresch and Tamvakis, establishing an upper bound of 1 for the absolute value of a specific hypergeometric series for all relevant integer parameters.

Contribution

The paper provides a proof of the Kresch-Tamvakis conjecture using advanced mathematical tools like Leonard pairs and the Perron-Frobenius theorem, which was previously unresolved.

Findings

The absolute value of the hypergeometric series is at most 1 for all valid parameters.

The proof employs the Biedenharn-Elliott identity and Leonard pairs.

The result confirms a conjecture in hypergeometric series theory.

Abstract

In this paper we resolve a conjecture of Kresch and Tamvakis. Our result is the following. Theorem: For any positive integer $D$ and any integers $i,j$ $(0\leq i,j \leq D)$, the absolute value of the following hypergeometric series is at most 1: \begin{equation*} {_4F_3} \left[ \begin{array}{c} -i, \; i+1, \; -j, \; j+1 \\ 1, \; D+2, \; -D \end{array} ; 1 \right]. \end{equation*} To prove this theorem, we use the Biedenharn-Elliott identity, the theory of Leonard pairs, and the Perron-Frobenius theorem.