Expansions for (q)∞n2+2n and basic hypergeometric series in U(n) Original Research Article

In this paper we derive new, more symmetrical expansions for (q; q)∞n2+2n by means of our multivariable generalization of Andrewsʹ variation of the standard proof of Jacobiʹs (q; q)∞3 result. Our proof relies upon a new multivariable extension of the Jacobi triple product identity. This result is deduced elsewhere by the second author from a U(n) multiple basic hypergeometric series generalization of Watsonʹs very-well-poised 8φ7 transformation. The derivation of our (q; q)∞n2+2n result utilizes partial derivatives and dihedral group symmetries to write the sum over regions in n-space. In addition, we prove that our expansions for (q; q)∞n2+2n are equivalent to Macdonaldʹs An family of eta-function identities.