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We study the vector space V_k^m(\lambda) of shifted polyharmonic Maass formsof weight k \in 2Z, depth m \geq 0, and shift \lambda \in C. This space iscomposed of real-analytic modular forms of weight k for PSL(2,Z) with moderategrowth at the cusp which are annihilated by (\Delta_k - \lambda)^m, where\Delta_k is the weight k hyperbolic Laplacian. We treat the case \lambda \neq0, complementing work of the second and third authors on polyharmonic Maassforms (with no shift). We show that V_k^m(\lambda) is finite-dimensional andbound its dimension. We explain the role of the real-analytic Eisenstein seriesE_k(z,s) with \lambda=s(s+k-1) and of the differential operator d/ds in thistheory.

Shifted polyharmonic Maass forms for PSL(2,Z)