Expansions for \((q)_\infty^{n^2+2n}\) and basic hypergeometric series in \(U(n)\)
type: Item
instance of: scholarly article
title: Expansions for \((q)_\infty^{n^2+2n}\) and basic hypergeometric series in \(U(n)\)
zbMATH Open document ID: 0936.33010
DOI: 10.1016/S0012-365X(98)00375-6
author: Verne E. Leininger
author: Stephen C. Milne
published in: Discrete Mathematics
publication date: 1999-11-23T00:00:00Z
review text: 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.
reviewed by: W. Koepf
Mathematics Subject Classification ID: 33D70
Mathematics Subject Classification ID: 05E99
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 1331477
zbMATH Keywords: Multiple basic hypergeometric series
zbMATH Keywords: Unitary groups \(U(n)\)
zbMATH Keywords: \(q\)-Whipple transformations
zbMATH Keywords: \(\eta\)-functions
zbMATH Keywords: Macdonald identities
zbMATH Keywords: dihedral group symmetries
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1331477