Heine's basic transform and a permutation group for q-harmonic series

More than 150 years ago, E. Heine considered the series [formula unable to be reproduced here] and proved several results for it. Today generalized Heine series become an actively investigated part of modern mathematics and a lot of papers and monographs are devoted to their study. The aim of this note is to make use of Heine's transform (1) to deduce a sharp irrationality measure for the q-harmonic series.