chapter
Theorem 12.4 (Dougall’s Formula).
Pages 1

Theorem 12.4 (Dougall’s Formula). For any n ∈ N and a, b, c, d, e ∈ C satisfying the condition

b + c + d + e = n− 1, (12.11)

we have

( n

k

) [n + 2a]k[b + a]k[c + a]k[d + a]k[e + a]k[n− 2a]n−k

× [b− a]n−k[c− a]n−k[d− a]n−k[e− a]n−k(n + 2a− 2k) = [n + 2a]2n+1[b + e]n[c + e]n[d + e]n = (−1)n[n + 2a]2n+1[b + c]n[c + d]n[d + b]n. (12.12)

Proof: Condition (12.11) implies that we can write

[n− b− c− 1]n−j = [d + e]n−j ,

and as we have

[n− d− e− 1]j = [d + e− n + j]j(−1)j ,

the product of these two factors from the right sum of (12.7) becomes

[n−b−c−1]n−j [n−d−e−1]j = [d+e]n−j [d+e−n+j]j(−1)j = [d+e]n(−1)j .

The right sum therefore becomes