This is a survey of results in \(q\)-series with direct or possible applications to the study of quantum groups. The emphasis is on \(q\)-commuting variables, \(x\) and \(y\), that satisfy either \(xy = qyx\) or the more general \(q\)-Heisenberg relation: \(xy-qyx = (1-q)c\) where \(c\) commutes with \(x\) and \(y\). Koornwinder discusses Schützenberger’s \(q\)-binomial formula, various functional equations for \(q\)-exponentials and \(q\)-logarithms, and the reformulation of some of these identities in terms of commuting variables. Inspired by a paper of A. Kempf and S. Majid [J. Math. Phys. 35, No. 12, 6802-6837 (1994; Zbl 0826.17018)], he presents results on translation invariance of Jackson’s \(q\)-integral and a \(q\)-Fourier transform pair connected to discrete \(q\)-Hermite polynomials. The paper concludes with a description of the relationship of these topics to Majid’s braided quantum groups.
For the entire collection see [Zbl 0864.00058].