Statements
The original identity, from , is : A generalization, also sometimes called Dixon's identity, is : where ''a'', ''b'', and ''c'' are non-negative integers . The sum on the left can be written as the terminating well-poised hypergeometric series : and the identity follows as a limiting case (as ''a'' tends to an integer) of Dixon's theorem evaluating a well-poised 3''F''2 generalized hypergeometric series at 1, from : : This holds for Re(1 + ''a'' − ''b'' − ''c'') > 0. As ''c'' tends to −∞ it reduces to Kummer's formula for the hypergeometric function 2F1 at −1. Dixon's theorem can be deduced from the evaluation of the Selberg integral.''q''-analogues
A ''q''-analogue of Dixon's formula for the basic hypergeometric series in terms of the q-Pochhammer symbol is given by : where , ''qa''1/2/''bc'', < 1.References
* * * * * * {{citation , last=Wilf , first=Herbert S. , authorlink=Herbert Wilf , title=Generatingfunctionology , edition=2nd , location=Boston, MA , publisher=Academic Press , year=1994 , isbn=0-12-751956-4 , zbl=0831.05001 Enumerative combinatorics Factorial and binomia