Saalschütz's Theorem

(1)

where  is a generalized hypergeometric function and  is the gamma function. It can be derived from the Dougall-Ramanujan identity and written in the symmetric form

(2)

for

(3)

with  a nonpositive integer and  the Pochhammer symbol (Bailey 1935, p. 9; Petkovšek et al. 1996; Koepf 1998, p. 32). If one of , , and  is nonpositive but it is not known which, an alternative formulation due to W. Gosper (pers. comm.) gives the form

(4)

which is symmetric in  and .

If instead

(5)

then

(6)

(W. Gosper, pers. comm.).

REFERENCES:

Bailey, W. N. "Saalschütz's Theorem." §2.2 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, p. 9, 1935.

Dougall, J. "On Vandermonde's Theorem and Some More General Expansions." Proc. Edinburgh Math. Soc. 25, 114-132, 1907.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, p. 104, 1999.

Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.

Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, pp. 43 and 126, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.

Saalschütz, L. "Eine Summationsformel." Z. für Math. u. Phys. 35, 186-188, 1890.

Saalschütz, L. "Über einen Spezialfall der hypergeometrischen Reihe dritter Ordnung." Z. für Math. u. Phys. 36, 278-295 and 321-327, 1891.

Shepard, W. F. "Summation of the Coefficients of Some Terminating Hypergeometric Series." Proc. London Math. Soc. 10, 469-478, 1912.