Some new infinite families of \(\eta\)-function identities
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 similar lines, the authors also derive three infinite families of expansions for the products of powers of \((q^a;q^m)_\infty\). Also included in the paper is a \(U(n)\) extension of the quintuple product identity. In the final section, the authors answer completely the question for which exponents \(c\) there are expansions for \(\eta(q)^c\) by the authors \textit{and} by Macdonald. Since this boils down to comparing the exponents that appear in the authors' work with those in Macdonald's, this requires to solve certain quadratic diophantine equations in three variables.
very-well-poised on unitary groups \(U(n+1)\)
scientific article; zbMATH DE number 1513051