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Dougall-Ramanujan Identity
A hypergeometric identity discovered by Ramanujan around From Hardy (, pp. 13 and ),
(1) |
where
(2) |
is the rising factorial (a.k.a.
Pochhammer symbol,
(3) |
is the falling factorial (Hardy , p. ), is a gamma function, and one of
(4) |
is a positive integer.
Equation (1) can also be rewritten as
(5) |
(Hardy , p. ). In a more symmetric form, if , , , and for , 2, , 6, then
(6) |
where is the Pochhammer symbol (Petkovšek et al.
Dougall ramanujan biography
).
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.