Traces of CM values and cycle integrals of polyharmonic Maass forms

Lagarias and Rhoades generalized harmonic Maass forms by considering forms which are annihilated by a number of iterations of the action of the ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document}-operator. In our previous work, we considered polyharmonic weak Maass forms by allowing the exponential growth at cusps, and constructed a basis of the space of such forms. This paper focuses on the case of half-integral weight. We construct a basis as an analogue of our work, and give arithmetic formulas for the Fourier coefficients in terms of traces of CM values and cycle integrals of polyharmonic weak Maass forms. These results put the known results into a common framework.