Another simple proof of the quintuple product identity

The quintuple identity has a long history and, as Berndt [5] points out, it is difficult to assign priority to it. It seems that a proof of the identity was first published in H. A. Schwartz’s book in 1893 [19]. Watson gave a proof in 1929 in his work on the RogersRamanujan continued fractions [20]. Since then, various proofs have appeared. To name a few, Carlitz and Subbarao gave a simple proof in [8]; Andrews [2] gave a proof involving basic hypergeometric functions; Blecksmith, Brillhart, and Gerst [7] pointed out that the quintuple identity is a special case of their theorem; and Evans [11] gave a short and elegant proof by using complex function theory. For updated history up to the late 80s and early 90s, see Hirschhorn [15] (in which the author also gave a beautiful generalization of the quintuple identity) and Berndt [5] (in which the author also gave a proof that ties the quintuple identity to the larger framework of the work of Ramanujan on q-series and theta functions; see also [1]). Since the early 90s, several authors gave different new proofs of the quintuple identity; see [6, 13, 12, 17]. See also Cooper’s papers [9, 10] for the connections between the quintuple product identity and Macdonald identities [18]. Quite recently, Kongsiriwong and Liu [16] gave an interesting proof that makes use of the cube root of unity.