A Summation Theorem for Hypergeometric Series Very-Well-Poised on $G_2 $

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An analogue of Bailey’s ${}_6 \psi _6 $ summation theorem is proved for basic hypergeometric series that are very well poised on the Lie algebra $G_2 $. As a limiting case, a new proof of the Macdonald identity associated to the affine root system of type $G_2 $ is obtained. A summation theorem for ordinary hypergeometric series that are very well poised on $G_2 $ is proved by Carlson’s theorem.