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Saalschütz's Theorem


Saalschütz's theorem is the generalized hypergeometric function identity

 _3F_2[a,b,-n; c,1+a+b-c-n;1]=((c-a)_n(c-b)_n)/((c)_n(c-a-b)_n)
(1)

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

The identity is sometimes also written

 _3F_2[a,b,c; d,e;1]=((d-a)_(|c|)(d-b)_(|c|))/((d)_(|c|)(d-a-b)_(|c|))
(2)

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

 a+b+c+1=d+e
(3)

and c is a nonpositive integer.

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

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

 a+b+c+1=d+e
(4)

which is symmetric in (a,b,c) and (d,e) can be given by

 _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))
(5)

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

If instead

 a+b+c+2=d+e,
(6)

then

 _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))
(7)

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


See also

Dougall-Ramanujan Identity, Generalized Hypergeometric Function, Kummer's Theorem

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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, 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.

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Saalschütz's Theorem

Cite this as:

Weisstein, Eric W. "Saalschütz's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SaalschuetzsTheorem.html

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