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Shifted polyharmonic Maass forms for $$\text{PSL} (2,{\mathbb Z})$$. (English) Zbl 1440.11078
Summary: We study the vector space $$V_k^m(\lambda)$$ of shifted polyharmonic Maass forms of weight $$k\in 2\mathbb Z$$, depth $$m\geq 0$$, and shift $$\lambda\in \mathbb C$$. This space is composed of real-analytic modular forms of weight $$k$$ for $$\operatorname{PSL}(2,\mathbb Z)$$ with moderate growth at the cusp which are annihilated by $$(\varDelta_k - \lambda)^m$$, where $$\varDelta_k$$ is the weight $$k$$ hyperbolic Laplacian. We treat the case $$\lambda \neq 0$$, complementing work of the second and third authors on polyharmonic Maass forms (with no shift). We show that $$V_k^m(\lambda)$$ is finite-dimensional and bound its dimension. We explain the role of the real-analytic Eisenstein series $$E_k(z,s)$$ with $$\lambda=s(s+k-1)$$ and of the differential operator $$\frac{\partial}{\partial s}$$ in this theory.

##### MSC:
 11F55 Other groups and their modular and automorphic forms (several variables) 11F37 Forms of half-integer weight; nonholomorphic modular forms 11F12 Automorphic forms, one variable
##### Keywords:
modular forms; polyharmonic; harmonic; Maass forms
DLMF
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##### References:
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