Proof of the Kresch-Tamvakis conjecture

In this paper we resolve a conjecture of Kresch and Tamvakis [Duke Math. J. 110 (2001), pp. 359–376]. Our result is the following. Theorem : For any positive integer D D and any integers i , j i,j ( 0 ≤ i , j ≤ D ) , (0\leq i,j \leq D), \; the absolute value of the following hypergeometric series is at most 1: 4 F 3 [ − i , i + 1 , − j , j + 1 1 , D + 2 , − D ; 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.