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Dougall-Ramanujan Identity

A hypergeometric identity discovered by Ramanujan around 1910. From Hardy (1999, pp. 13 and 102-103),

sum_(n=0)^infty(-1)^n(s+2n)(s^((n))(x+y+z+u+2s+1)^((n)))/((x+y+z+u-s)_((n)))product_(x,y,z,u)(x_((n)))/((x+s+1)^((n))) =s/(Gamma(s+1)Gamma(x+y+z+u+s+1))product_(x,y,z,u)(Gamma(x+s+1)Gamma(y+z+u+s+1))/(Gamma(z+u+s+1)),
(1)

where

a^((n))=a(a+1)...(a+n-1)
(2)

is the rising factorial (a.k.a. Pochhammer symbol,

a_((n))=a(a-1)...(a-n+1)
(3)

is the falling factorial (Hardy 1999, p. 101), Gamma(z) is a gamma function, and one of

x,y,z,u,-x-y-z-u-2s-1
(4)

is a positive integer.

Equation (1) can also be rewritten as

_7F_6[s,1+1/2s,-x,-y,-z,-u,x-y+z+u+2s+1 ; 1/2s,x+s+1,y+s+1,z+s+1,u+s+1, ; -x-y-z-u-s;1] =1/(Gamma(s+1)Gamma(x+y+z+u+s+1))product_(x,y,z,u)(Gamma(x+s+1)Gamma(y+z+u+s+1))/(Gamma(z+u+s+1)).
(5)

(Hardy 1999, p. 102). In a more symmetric form, if n=2a_1+1=a_2+a_3+a_4+a_5, a_6=1+a_1/2, a_7=-n, and b_i=1+a_1-a_(i+1) for i=1, 2, ..., 6, then

_7F_6[a_1,a_2,a_3,a_4,a_5,a_6,a_7; b_1,b_2,b_3,b_4,b_5,b_6;1] =((a_1+1)_n(a_1-a_2-a_3+1)_n)/((a_1-a_2+1)_n(a_1-a_3+1)_n)((a_1-a_2-a_4+1)_n(a_1-a_3-a_4+1)_n)/((a_1-a_4+1)_n(a_1-a_2-a_3-a_4+1)_n),
(6)

where (a)_n is the Pochhammer symbol (Petkovšek et al. 1996).

The identity is a special case of Jackson's identity, and gives Dixon's theorem, Saalschütz's theorem, and Morley's formula as special cases.

See also

Bailey's Transformation, Dixon's Theorem, Dougall's Theorem, Generalized Hypergeometric Function, Hypergeometric Function, Jackson's Identity, Morley's Formula, Rogers-Ramanujan Identities, Saalschütz's Theorem

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References

Bailey, W. N. "An Elementary Proof of Dougall's Theorem." §5.1 in Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, pp. 25-26 and 34, 1935.Dixon, A. C. "Summation of a Certain Series." Proc. London Math. Soc. 35, 285-289, 1903.Dougall, J. "On Vandermonde's Theorem and Some More General Expansions." Proc. Edinburgh Math. Soc. 25, 114-132, 1907.Hardy, G. H. "A Chapter from Ramanujan's Note-Book." Proc. Cambridge Philos. Soc. 21, 492-503, 1923.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, pp. 43, 126-127, and 183-184, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.

Referenced on Wolfram|Alpha

Dougall-Ramanujan Identity

Cite this as:

Weisstein, Eric W. "Dougall-Ramanujan Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Dougall-RamanujanIdentity.html

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