Some new infinite families of \(\eta\)-function identities
type: Item
instance of: scholarly article
title: Some new infinite families of \(\eta\)-function identities
zbMATH Open document ID: 0968.33008
author: Verne E. Leininger
author: Stephen C. Milne
published in: Methods and Applications of Analysis
publication date: 2001-09-12T00:00:00Z
review text: The famous Dedekind \(\eta\)-function is given by \(\eta(q)=q^{1/24}(q;q)_\infty\), where \((q;q)_\infty=\prod _{i=1} ^{\infty}(1-q^i)\). The problem of finding expansions for powers of \(\eta\)-functions has always attracted mathematicians. The first one found of such expansions are Euler's expansion for the first power \(\eta(q)\) and Jacobi's expansion for the third power \(\eta(q)^3\). Expansions \(\eta(q)^c\) for many more particular values of \(c\) are now known. In particular, in his landmark paper [Invent. Math. 15, 91-143 (1972; Zbl 0244.17005)], \textit{I. G. Macdonald} showed that almost all of these expansions come from affine root systems. The authors of the present paper derive new expansions for two infinite families of powers of \(\eta\)-functions, namely for \(\eta(q)^{n^2+2}\) and \(\eta(q)^{n^2}\), where \(n\) is any positive integer. They also present a conjectured expansion for \(\eta(q)^{n^2-2}\). In a previous paper [Discrete Math. 204, 281-317 (1999; Zbl 0936.33010)], the authors had provided expansions for \(\eta(q)^{n^2+2n}\). All these expansions are even new in the cases when the power of \(\eta(q)\) is such that one of Macdonald's identities applies (i.e., in those cases, the authors' and Macdonald's expansions are intrinsically different). The authors prove their result by following their recipe in [loc. cit.]: they start with a summation identity for a \(U(n)\) basic hypergeometric series due to the second author, and apply differentiation with respect to the appearing parameters together with subsequent specialization of them in such a way that the product side of the summation identity becomes a power of \((q;q)_\infty\). The proof is finished off by simplifying the resulting sum side by the use of a well-known generalised form of the Vandermonde determinant identity, and by multiplying both sides of the resulting identity by the appropriate power of \(q\) in order to convert \((q;q)_\infty\) into \(\eta(q)\). By following along simi…
reviewed by: C. Krattenthaler
Mathematics Subject Classification ID: 33D70
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 05E99
zbMATH DE Number: 1513051
zbMATH Keywords: multiple basic hypergeometric series
zbMATH Keywords: very-well-poised on unitary groups \(U(n+1)\)
zbMATH Keywords: quintuple product identity
zbMATH Keywords: Dedekind eta function
zbMATH Keywords: Macdonald identities
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: Pell's equation
zbMATH Keywords: Pell numbers
zbMATH Keywords: Pythagorean triangles
zbMATH Keywords: Lucas sequences
zbMATH Keywords: identities of Rogers-Ramanujan type
MaRDI profile type: Publication
full work available at URL: maa.1999.v6.n2.a7
OpenAlex ID: W50304216
DOI: 10.4310/MAA.1999.V6.N2.A7
description: scientific article; zbMATH DE number 1513051