Result: Expansions for (q)∞n2+2n and basic hypergeometric series in U(n): Expansions for \((q)_\infty^{n^2+2n}\) and basic hypergeometric series in \(U(n)\)

In this paper the authors derive new, more symmetrical expansions for \((q;q)^{n^{2}+2n}_\infty\) by means of their multivariable generalization of Andrews' variation of the standard proof of Jacobi's \((q;q)^3_\infty\) result. Their 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 Watson's very-well-poised \(_8\phi_7\) transformation. The derivation of their \((q;q)^{n^{2}+2n}_\infty\) result utilizes partial derivatives and dihedral group symmetries to write the sum over regions in \(n\)-space. In addition, the authors prove that their expansions for \((q;q)^{n^{2}+2n}_\infty\) are equivalent to Macdonald's \(A_n\) family of \(\eta\)-function identities.