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.