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, p. 32, 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.
![_3F_2[a,b,-n; c,1+a+b-c-n;1]=((c-a)_n(c-b)_n)/((c)_n(c-a-b)_n)](/images/equations/SaalschuetzsTheorem/NumberedEquation1.svg)
非負整數成立，其中 
是 波赫哈默爾符號（Saalschütz 1890；Bailey 1935，第 9 頁）。![_3F_2[a,b,c; d,e;1]=((d-a)_(|c|)(d-b)_(|c|))/((d)_(|c|)(d-a-b)_(|c|))](/images/equations/SaalschuetzsTheorem/NumberedEquation2.svg)
是 非正整數。
、
和 
中的一個或兩個是非正整數，則可以使用定義的形式
和 
中是對稱的，可以由下式給出![_3F_2[a,b,c; d,e;1]=(Gamma(d))/(Gamma(d-a)Gamma(d-b)Gamma(d-c))(Gamma(e))/(Gamma(e-a)Gamma(e-b)Gamma(e-c))(pi^2)/(cos(pid)cos(pie)+cos(pia)cos(pib)cos(pic))](/images/equations/SaalschuetzsTheorem/NumberedEquation5.svg)
![_3F_2[a,b,c; d,e;1]=pi^2(Gamma(d))/(Gamma(d-a)Gamma(d-b)Gamma(d-c))(Gamma(e))/(Gamma(e-a)Gamma(e-b)Gamma(e-c))(de-(a+1)(b+1)(c+1)+abc)/(cos(pid)cos(pie)-cos(pia)cos(pib)cos(pic))](/images/equations/SaalschuetzsTheorem/NumberedEquation7.svg)
