Saalschütz's Theorem

Saalschütz's theorem is the generalized hypergeometric function identity

(1)

which holds for a nonnegative integer and where is a Pochhammer symbol (Saalschütz 1890; Bailey 1935, p. 9).

The identity is sometimes also written

(2)

(Bailey 1935, p. 9; Petkovšek et al. 1996, p. 43), where

(3)

and is a nonpositive integer.

Saalschütz's theorem can be derived from the Dougall-Ramanujan identity.

If one or two of , , and are nonpositive integers, a formulation making use of the definition

(4)

which is symmetric in and can be given by

(5)

(W. Gosper, pers. comm., likely some time in the late 1990s).

If instead

(6)

then

(7)

(W. Gosper, pers. comm, likely some time in the late 1990s).