Modular forms of real weights and generalised Dedekind symbols

In a previous paper, I have defined non-commutative generalised Dedekind symbols for classical PSL(2,Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$PSL(2,\mathbf {Z})$$\end{document}-cusp forms using iterated period polynomials. Here I generalise this construction to forms of real weights using their iterated period functions introduced and studied in a recent article by R. Bruggeman and Y. Choie.