Proof of the Kresch-Tamvakis Conjecture

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 $i,j$ $(0\le i,j\le D)$, 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.