Polyharmonic weak Maass forms of higher depth for SL_2(Z)

The space of polyharmonic Maass forms was introduced by Lagarias-Rhoades, recently. They constructed its basis from the Taylor coefficients of the real analytic Eisenstein series. In this paper, we introduce polyharmonic weak Maass forms, that is, we relax the moderate growth condition at cusp, and we construct a basis as a generalization of Lagarias-Rhoades’ works. As a corollary, we can obtain a preimage of an arbitrary polyharmonic weak Maass form under the xi-operator.