Dougall-Ramanujan Identity

Discovered by Ramanujan around 1910. From Hardy (1959, pp. 102-103),

$\sum_{n=0}^\infty (-1)^n(s+2n){s^{(n)}\over 1^{(n)}} {(x+y+z+u+2s+1)^{(n)}\over (x+y+z+u-s)_{(n)}}\prod_{x,y,z,u} {x_{(n)}\over (x+s+1)^{(n)}}$
$= {s\over \Gamma(s+1)\Gamma(x+y+z+u+s+1)}\prod_{x,y,z,u} {\Gamma(x+s+1)\Gamma(y+z+u+s+1)\over \Gamma(z+u+s+1)},\quad$	(1)

where

$$a^{(n)}\equiv a(a+1)\cdots (a+n-1)$$ (2)

$$a_{(n)}\equiv a(a-1)\cdots (a-n+1)$$ (3)

(here, the Pochhammer Symbol has been written $a^{(n)}$). This can be rewritten as

${}_7F_6\left[{\matrix{s,1+{\textstyle{1\over 2}}s,-x -y,-z,-u,x-y+z+u+2s+1\hfil... ...e{1\over 2}}s, x+s+1, y+s+1, z+s+1, u+s+1,\hfil\cr \hfil -x-y-z-u-s}; 1}\right]$
$= {1\over\Gamma(s+1)\Gamma(x+y+z+u+s+1)}\prod_{x, t, z, u}{\Gamma(x+s+1)\Gamma(y+z+u+s+1)\over\Gamma(z+u+s+1)}.\quad$	(4)

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

${}_7F_6\left[{\matrix{a_1, a_2, a_3, a_4, a_5, a_6, a_7\cr \hfil b_1, b_2, b_3, b_4, b_5, b_6\cr}; 1}\right]$
$={(a_1+1)_n(a_1-a_2-a_3+1)_n\over (a_1-a_2+1)_n(a_1-a_3+1)_n}{(a_1-a_2-a_4+1)_n(a_1-a_3-a_4+1)_n\over (a_1-a_4+1)_n(a_1-a_2-a_3-a_4+1)_n},\quad$	(5)

where $(a)_n$ is the Pochhammer Symbol (Petkovsek et al. 1996).

The identity is a special case of Jackson's Identity.

See also Dixon's Theorem, Dougall's Theorem, Generalized Hypergeometric Function, Hypergeometric Function, Jackson's Identity, Saalschütz's Theorem

References

Dixon, A. C. ``Summation of a Certain Series.'' Proc. London Math. Soc. 35, 285-289, 1903.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1959.

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, pp. 43, 126-127, and 183-184, 1996.