Proof: Substitute p = 1 and p = 2 in (9.26) and p = −1 in (9.27) and eventually apply the Chu-Vandermonde formula (8.3) to each term.
Remarks 9.12. Surprisingly, Sn(n+2a, b+a, b−a, n−2a−1) is symmetric in (a, b). Another symmetry for the balanced Dixon formula is
Sn(n + 2a, b + a,b− a, n− 2a) = Sn (n− 2b− 1,−b− a− 1, b− a, n + 2b + 1) . (9.31)
Furthermore, it does not matter which term is changed by the deviation p. If we apply (9.6) to Sn(n+2a, b+a, b−a, n−2a−p), we get the symmetric expression
) [−a− b− 1]k[a + b]k[b− a]n−k[−1 + a− b + p]n−k(−1)k.