These identities famously follow from the MacMahon Master theorem, and can now be routinely proved by computer algorithms (Ekhad 1990).
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 3F2 generalized hypergeometric series at 1, from (Dixon 1902): This holds for Re(1 + 1⁄2a − 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.
A q-analogue of Dixon's formula for the basic hypergeometric series in terms of the q-Pochhammer symbol is given by where |qa1/2/bc| < 1.